2001
2001
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1.I.1C
Part IA, 2001 commentShow, using the summation convention or otherwise, that (a.b)c, for a, b, c
The function is defined by where is a unit vector in . Show that is linear and find the elements of a matrix such that for all .
Find all solutions to the equation . Evaluate . Describe the function П geometrically. Justify your answer.
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1.I.2C
Part IA, 2001 commentDefine what is meant by the statement that the vectors are linearly independent. Determine whether the following vectors are linearly independent and justify your answer.
For the vectors taken from a real vector space consider the statements A) are linearly dependent, B) , C) , not all , D) , not both , E) , F) basis of that contains all 3 vectors .
State if the following implications are true or false (no justification is required): i) , vi) , ii) , vii) , iii) , viii) , iv) , ix) , v) , x) .
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1.II.5C
Part IA, 2001 commentThe matrix
defines a linear map by . Find a basis for the kernel of for all values of .
Let and be bases of . Show that there exists a matrix , to be determined in terms of and , such that, for every linear mapping , if has matrix with respect to and matrix with respect to , then .
For the bases
find the basis transformation matrix and calculate .
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1.II.6C
Part IA, 2001 commentAssume that is a particular solution to the equation with and a real matrix . Explain why the general solution to is given by where is any vector such that .
Now assume that is a real symmetric matrix with three different eigenvalues and . Show that eigenvectors of with respect to different eigenvalues are orthogonal. Let be a normalised eigenvector of with respect to the eigenvalue , . Show that the linear system
where denotes the unit matrix, is solvable if and only if . Show that the general solution is given by
[Hint: consider the components of and with respect to a basis of eigenvectors of .]
Consider the matrix and the vector
Verify that and are eigenvectors of . Show that is solvable and find its general solution.
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1.II.7C
Part IA, 2001 commentFor and the equation describes a circle in the complex plane. Find its centre and radius. What does the equation describe if ? Sketch the circles for and .
Show that the complex function for satisfies .
[Hint: means that and such that
For two circles and a function is defined by
Prove that . Show that
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1.II.8C
Part IA, 2001 commentLet denote the straight line through with directional vector
Show that is a subspace of and show that for some .
For fixed let be the set of all the parallel straight lines with directional vector . On an addition and a scalar multiplication are defined by
Explain why these operations are well-defined. Show that the addition is associative and that there exists a zero vector which should be identified.
You may now assume that is a vector space. If is a basis for show that is a basis for .
For a linear map is defined by
Find the matrix of with respect to the basis .
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3.I.1F
Part IA, 2001 commentFor a matrix , prove that if and only if and . Prove that if and only if .
[Hint: it is easy to check that
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3.I.2D
Part IA, 2001 commentShow that the set of Möbius transformations of the extended complex plane form a group. Show further that an arbitrary Möbius transformation can be expressed as the composition of maps of the form
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3.II.5F
Part IA, 2001 commentLet be matrices, real or complex. Define the trace to be the sum of diagonal entries . Define the commutator to be the difference . Give the definition of the eigenvalues of a matrix and prove that it can have at most two distinct eigenvalues. Prove that a) , b) equals the sum of the eigenvalues of , c) if all eigenvalues of are equal to 0 then , d) either is a diagonalisable matrix or the square , e) where and is the unit matrix.
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Part IA, 2001
commentDefine the notion of an action of a group on a set . Define orbit and stabilizer, and then, assuming that is finite, state and prove the Orbit-Stabilizer Theorem.
Show that the group of rotations of a cube has order 24 .
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3.II.7E
Part IA, 2001 commentState Lagrange's theorem. Use it to describe all groups of order , where is a fixed prime number.
Find all the subgroups of a fixed cyclic group of order .
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3.II.8D
Part IA, 2001 comment(i) Let denote the alternating group of even permutations of four symbols. Let be the 3-cycle and be the pairs of transpositions and . Find , and show that is generated by and .
(ii) Let and be groups and let
Show how to make into a group in such a way that contains subgroups isomorphic to and .
If is the dihedral group of order and is the cyclic group of order 2 , show that is isomorphic to . Is the group isomorphic to ?
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1.I
Part IA, 2001 commentWhat does it mean to say that as ?
Show that, if and , then as .
If further for all and , show that as .
Give an example to show that the non-vanishing of for all need not imply the non-vanishing of .
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1.I.4D
Part IA, 2001 commentStarting from the theorem that any continuous function on a closed and bounded interval attains a maximum value, prove Rolle's Theorem. Deduce the Mean Value Theorem.
Let be a differentiable function. If for all show that is a strictly increasing function.
Conversely, if is strictly increasing, is for all ?
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1.II.9D
Part IA, 2001 comment(i) If are complex numbers show that if, for some , the set is bounded and , then converges absolutely. Use this result to define the radius of convergence of the power series .
(ii) If as show that has radius of convergence equal to .
(iii) Give examples of power series with radii of convergence 1 such that (a) the series converges at all points of the circle of convergence, (b) diverges at all points of the circle of convergence, and (c) neither of these occurs.
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1.II.10D
Part IA, 2001 commentSuppose that is a continuous real-valued function on with . If show that there exists with and .
Deduce that if is a continuous function from the closed bounded interval to itself, there exists at least one fixed point, i.e., a number belonging to with . Does this fixed point property remain true if is a continuous function defined (i) on the open interval and (ii) on ? Justify your answers.
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1.II.11D
Part IA, 2001 comment(i) Show that if is twice continuously differentiable then, given , we can find some constant and such that
for all .
(ii) Let be twice continuously differentiable on (with one-sided derivatives at the end points), let and be strictly positive functions and let .
If and a sequence is defined by , show that is a decreasing sequence of points in and hence has limit . What is ? Using part (i) or otherwise estimate the rate of convergence of to , i.e., the behaviour of the absolute value of for large values of .
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1.II.12D
Part IA, 2001 commentExplain what it means for a function to be Riemann integrable on , and give an example of a bounded function that is not Riemann integrable.
Show each of the following statements is true for continuous functions , but false for general Riemann integrable functions .
(i) If is such that for all in and , then for all in .
(ii) is differentiable and .
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2.I.1B
Part IA, 2001 commentFind the solution to
in the range subject to , where is the Heavyside function defined by
Sketch the solution.
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2.I.2B
Part IA, 2001 commentThe function satisfies the inhomogeneous second-order linear differential equation
Find the solution that satisfies the conditions that and is bounded as .
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2.II.5B
Part IA, 2001 commentThe real sequence satisfies the difference equation
Show that the general solution can be written
where and are arbitrary real constants.
Now let satisfy
Show that a particular solution of can be written in the form
where
and .
Hence, find the general solution to .
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Part IA, 2001
commentThe function satisfies the linear equation
The Wronskian, , of two independent solutions denoted and is defined to be
Let be given. In this case, show that the expression for can be interpreted as a first-order inhomogeneous differential equation for . Hence, by explicit derivation, show that may be expressed as
where the rôle of should be briefly elucidated.
Show that satisfies
Verify that is a solution of
Hence, using with and expanding the integrand in powers of to order , find the first three non-zero terms in the power series expansion for a solution, , of ( ) that is independent of and satisfies .
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2.II.7B
Part IA, 2001 commentConsider the linear system
where
where is real and is a real constant, .
Find a (complex) eigenvector, e, of and its corresponding (complex) eigenvalue, . Show that the second eigenvector and corresponding eigenvalue are respectively and , where the bar over the symbols signifies complex conjugation. Hence explain how the general solution to can be written as
where is complex.
Write down a differential equation for and hence, for , deduce the solution to which satisfies the initial condition .
Is the linear system resonant?
By taking the limit of the solution already found deduce the solution satisfying when .
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2.II.8B
Part IA, 2001 commentCarnivorous hunters of population prey on vegetarians of population . In the absence of hunters the prey will increase in number until their population is limited by the availability of food. In the absence of prey the hunters will eventually die out. The equations governing the evolution of the populations are
where and are positive constants, and and are non-negative functions of time, . By giving an interpretation of each term explain briefly how these equations model the system described.
Consider these equations for . In the two cases and determine the location and the stability properties of the critical points of . In both of these cases sketch the typical solution trajectories and briefly describe the ultimate fate of hunters and prey.
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4.I.3A
Part IA, 2001 commentDerive the equation
for the motion of a particle of mass under an attractive central force , where and is the distance of the particle from the centre of force, and where is the angular momentum of the particle about the centre of force.
[Hint: you may assume the expressions for the radial and transverse accelerations in the form .]
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4.I.4A
Part IA, 2001 commentTwo particles of masses and at positions and are subject to forces . Show that the centre of mass moves at a constant velocity. Obtain the equation of motion for the relative position of the particles. How does the reduced mass
of the system enter?
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4.II.9A
Part IA, 2001 commentThe position and velocity of a particle of mass are measured in a frame which rotates at constant angular velocity with respect to an inertial frame. Write down the equation of motion of the particle under a force .
Find the motion of the particle in coordinates with initial condition
where . Show that the particle has a maximum speed at , and find this speed.
[Hint: you may find it useful to consider the combination .]
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4.II.10A
Part IA, 2001 commentA spherical raindrop of radius and density falls down at a velocity through a fine stationary mist. As the raindrop falls its volume grows at the rate with constant . The raindrop is subject to the gravitational force and a resistive force with a positive constant. Show and satisfy
Find an expression for , and deduce that as time increases tends to the constant value , and thence the raindrop tends to a constant acceleration which is less than .
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4.II.11A
Part IA, 2001 commentA spacecraft of mass moves under the gravitational influence of the Sun of mass and with universal gravitation constant . After a disastrous manoeuvre, the unfortunate spacecraft finds itself exactly in a parabolic orbit about the Sun: the orbit with zero total energy. Using the conservation of energy and angular momentum, or otherwise, show that in the subsequent motion the distance of the spacecraft from the Sun satisfies
with constants and .
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4.II.12A
Part IA, 2001 commentFind the moment of inertia of a uniform solid cylinder of radius , length and total mass about its axis.
The cylinder is released from rest at the top of an inclined plane of length and inclination to the horizontal. The first time the plane is perfectly smooth and the cylinder slips down the plane without rotating. The experiment is then repeated after the plane has been roughened, so that the cylinder now rolls without slipping at the point of contact. Show that the time taken to roll down the roughened plane is times the time taken to slip down the smooth plane.
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4.I.1E
Part IA, 2001 comment(a) Show that, given a set , there is no bijection between and its power set.
(b) Does there exist a set whose members are precisely those sets that are not members of themselves? Justify your answer.
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4.I.2E
Part IA, 2001 commentProve, by induction or otherwise, that
Find the number of sequences consisting of zeroes and ones that contain exactly zeroes and at most ones.
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4.II.5E
Part IA, 2001 comment(a) Prove Wilson's theorem, that , where is prime.
(b) Suppose that is an odd prime. Express as a power of .
[Hint: .]
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4.II.6E
Part IA, 2001 commentState and prove the principle of inclusion-exclusion. Use it to calculate , where is Euler's -function.
In a certain large college, a survey revealed that of the fellows detest at least one of the pop stars Hairy, Dirty and Screamer. detest Hairy, detest Dirty and detest Screamer. If detest only Screamer and detest all three, what proportion detest Hairy and Dirty but not Screamer?
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4.II.7E
Part IA, 2001 comment(a) Prove that, if is prime and is not a multiple of , then .
(b) The order of is the least positive integer such that . Suppose now that ; what can you say about in terms of ? Show that .
(c) Suppose that is an odd prime. What is the order of if ? Find a condition on that is equivalent to the existence of an integer with .
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4.II.8E
Part IA, 2001 commentWhat is the Principle of Mathematical Induction? Derive it from the statement that every non-empty set of positive integers has a least element.
Prove, by induction on , that for all .
What is wrong with the following argument?
"Theorem: .
Proof: Assume that and . Add to both sides to get
So, by induction, the theorem is proved."
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2.I.3F
Part IA, 2001 commentThe following problem is known as Bertrand's paradox. A chord has been chosen at random in a circle of radius . Find the probability that it is longer than the side of the equilateral triangle inscribed in the circle. Consider three different cases:
a) the middle point of the chord is distributed uniformly inside the circle,
b) the two endpoints of the chord are independent and uniformly distributed over the circumference,
c) the distance between the middle point of the chord and the centre of the circle is uniformly distributed over the interval .
[Hint: drawing diagrams may help considerably.]
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2.I.4F
Part IA, 2001 commentThe Ruritanian authorities decided to pardon and release one out of three remaining inmates, and , kept in strict isolation in the notorious Alkazaf prison. The inmates know this, but can't guess who among them is the lucky one; the waiting is agonising. A sympathetic, but corrupted, prison guard approaches and offers to name, in exchange for a fee, another inmate (not who is doomed to stay. He says: "This reduces your chances to remain here from to : will it make you feel better?" hesitates but then accepts the offer; the guard names .
Assume that indeed will not be released. Determine the conditional probability
and thus check the guard's claim, in three cases:
a) when the guard is completely unbiased (i.e., names any of and with probability if the pair is to remain jailed),
b) if he hates and would certainly name him if is to remain jailed,
c) if he hates and would certainly name him if is to remain jailed.
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Part IA, 2001
commentI play tennis with my parents; the chances for me to win a game against are and against Dad , where . We agreed to have three games, and their order can be (where I play against Dad, then Mum then again Dad) or . The results of games are independent.
Calculate under each of the two orders the probabilities of the following events:
a) that I win at least one game,
b) that I win at least two games,
c) that I win at least two games in succession (i.e., games 1 and 2 or 2 and 3 , or 1 , 2 and 3,
d) that I win exactly two games in succession (i.e., games 1 and 2 or 2 and 3 , but not 1,2 and 3 ),
e) that I win exactly two games (i.e., 1 and 2 or 2 and 3 or 1 and 3 , but not 1,2 and 3.
In each case a)- e) determine which order of games maximizes the probability of the event. In case e) assume in addition that .
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2.II.10F
Part IA, 2001 commentA random point is distributed uniformly in a unit circle so that the probability that it falls within a subset is proportional to the area of . Let denote the distance between the point and the centre of the circle. Find the distribution function , the expected value and the variance .
Let be the angle formed by the radius through the random point and the horizontal line. Prove that and are independent random variables.
Consider a coordinate system where the origin is placed at the centre of . Let and denote the horizontal and vertical coordinates of the random point. Find the covariance and determine whether and are independent.
Calculate the sum of expected values . Show that it can be written as the expected value and determine the random variable .
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2.II.11F
Part IA, 2001 commentDipkomsky, a desperado in the wild West, is surrounded by an enemy gang and fighting tooth and nail for his survival. He has guns, , pointing in different directions and tries to use them in succession to give an impression that there are several defenders. When he turns to a subsequent gun and discovers that the gun is loaded he fires it with probability and moves to the next one. Otherwise, i.e. when the gun is unloaded, he loads it with probability or simply moves to the next gun with complementary probability . If he decides to load the gun he then fires it or not with probability and after that moves to the next gun anyway.
Initially, each gun had been loaded independently with probability . Show that if after each move this distribution is preserved, then . Calculate the expected value and variance Var of the number of loaded guns under this distribution.
[Hint: it may be helpful to represent as a sum of random variables taking values 0 and 1.]
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2.II.12F
Part IA, 2001 commentA taxi travels between four villages, , situated at the corners of a rectangle. The four roads connecting the villages follow the sides of the rectangle; the distance from to and to is 5 miles and from to and to miles. After delivering a customer the taxi waits until the next call then goes to pick up the new customer and takes him to his destination. The calls may come from any of the villages with probability and each customer goes to any other village with probability . Naturally, when travelling between a pair of adjacent corners of the rectangle, the taxi takes the straight route, otherwise (when it travels from to or to or vice versa) it does not matter. Distances within a given village are negligible. Let be the distance travelled to pick up and deliver a single customer. Find the probabilitites that takes each of its possible values. Find the expected value and the variance Var .
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3.I.3C
Part IA, 2001 commentFor a real function with and state the chain rule for the derivative .
By changing variables to and , where and with a suitable function to be determined, find the general solution of the equation
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3.I.4A
Part IA, 2001 commentSuppose that
Show that is an exact differential.
Show that
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3.II.9C
Part IA, 2001 commentExplain, with justification, how the nature of a critical (stationary) point of a function can be determined by consideration of the eigenvalues of the Hessian matrix of if is non-singular. What happens if is singular?
Let . Find the critical points of and determine their nature in the different cases that arise according to the values of the parameter .
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3.II.10A
Part IA, 2001 commentState the rule for changing variables in a double integral.
Let be the region defined by
Using the transformation and , show that
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3.II.11B
Part IA, 2001 commentState the divergence theorem for a vector field in a closed region bounded by a smooth surface .
Let be a scalar field. By choosing for arbitrary constant vector , show that
Let be the bounded region enclosed by the surface which consists of the cone with and the plane , where are cylindrical polar coordinates. Verify that holds for the scalar field where is a constant.
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3.II.12B
Part IA, 2001 commentIn show that, within a closed surface , there is at most one solution of Poisson's equation, , satisfying the boundary condition on
where and are functions of position on , and is everywhere non-negative.
Show that
are solutions of Laplace's equation on .
Find a solution of Laplace's equation in the region that satisfies the boundary conditions
where is a positive integer. Is your solution the only possible solution?
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1.I
Part IB, 2001 commentDetermine for which values of the matrix
is invertible. Determine the rank of as a function of . Find the adjugate and hence the inverse of for general .
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1.II.14C
Part IB, 2001 comment(a) Find a matrix over with both minimal polynomial and characteristic polynomial equal to . Furthermore find two matrices and over which have the same characteristic polynomial, , and the same minimal polynomial, , but which are not conjugate to one another. Is it possible to find a third such matrix, , neither conjugate to nor to ? Justify your answer.
(b) Suppose is an matrix over which has minimal polynomial of the form for distinct roots in . Show that the vector space on which defines an endomorphism decomposes as a direct sum into , where is the identity.
[Hint: Express in terms of and
Now suppose that has minimal polynomial for distinct . By induction or otherwise show that
Use this last statement to prove that an arbitrary matrix is diagonalizable if and only if all roots of its minimal polynomial lie in and have multiplicity
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2.I
Part IB, 2001 commentShow that right multiplication by defines a linear transformation . Find the matrix representing with respect to the basis
of . Prove that the characteristic polynomial of is equal to the square of the characteristic polynomial of , and that and have the same minimal polynomial.
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2.II.15C
Part IB, 2001 commentDefine the dual of a vector space . Given a basis of define its dual and show it is a basis of . For a linear transformation define the dual .
Explain (with proof) how the matrix representing with respect to given bases of and relates to the matrix representing with respect to the corresponding dual bases of and .
Prove that and have the same rank.
Suppose that is an invertible endomorphism. Prove that .
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3.I
Part IB, 2001 commentDetermine the dimension of the subspace of spanned by the vectors
Write down a matrix which defines a linear map whose image is and which contains in its kernel. What is the dimension of the space of all linear maps with in the kernel, and image contained in ?
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3.II.17C
Part IB, 2001 commentLet be a vector space over . Let be a nilpotent endomorphism of , i.e. for some positive integer . Prove that can be represented by a strictly upper-triangular matrix (with zeros along the diagonal). [You may wish to consider the subspaces for .]
Show that if is nilpotent, then where is the dimension of . Give an example of a matrix such that but .
Let be a nilpotent matrix and the identity matrix. Prove that has all eigenvalues equal to 1 . Is the same true of if and are nilpotent? Justify your answer.
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4.I
Part IB, 2001 commentFind the Jordan normal form of the matrix
and determine both the characteristic and the minimal polynomial of .
Find a basis of such that (the Jordan normal form of ) is the matrix representing the endomorphism in this basis. Give a change of basis matrix such that .
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4.II.15C
Part IB, 2001 commentLet and be matrices over . Show that and have the same characteristic polynomial. [Hint: Look at for , where and are scalar variables.]
Show by example that and need not have the same minimal polynomial.
Suppose that is diagonalizable, and let be its minimal polynomial. Show that the minimal polynomial of must divide . Using this and the first part of the question prove that and are conjugate.
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1.I.4B
Part IB, 2001 commentWrite down the Riemannian metric on the disc model of the hyperbolic plane. What are the geodesics passing through the origin? Show that the hyperbolic circle of radius centred on the origin is just the Euclidean circle centred on the origin with Euclidean radius .
Write down an isometry between the upper half-plane model of the hyperbolic plane and the disc model , under which corresponds to . Show that the hyperbolic circle of radius centred on in is a Euclidean circle with centre and of radius .
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1.II.13B
Part IB, 2001 commentDescribe geometrically the stereographic projection map from the unit sphere to the extended complex plane , and find a formula for . Show that any rotation of about the -axis corresponds to a Möbius transformation of . You are given that the rotation of defined by the matrix
corresponds under to a Möbius transformation of ; deduce that any rotation of about the -axis also corresponds to a Möbius transformation.
Suppose now that correspond under to distinct points , and let denote the angular distance from to on . Show that is the cross-ratio of the points , taken in some order (which you should specify). [You may assume that the cross-ratio is invariant under Möbius transformations.]
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3.I.4B
Part IB, 2001 commentState and prove the Gauss-Bonnet theorem for the area of a spherical triangle.
Suppose is a regular dodecahedron, with centre the origin. Explain how each face of gives rise to a spherical pentagon on the 2 -sphere . For each such spherical pentagon, calculate its angles and area.
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3.II.14B
Part IB, 2001 commentDescribe the hyperbolic lines in the upper half-plane model of the hyperbolic plane. The group acts on via Möbius transformations, which you may assume are isometries of . Show that acts transitively on the hyperbolic lines. Find explicit formulae for the reflection in the hyperbolic line in the cases (i) is a vertical line , and (ii) is the unit semi-circle with centre the origin. Verify that the composite of a reflection of type (ii) followed afterwards by one of type (i) is given by .
Suppose now that and are distinct hyperbolic lines in the hyperbolic plane, with denoting the corresponding reflections. By considering different models of the hyperbolic plane, or otherwise, show that
(a) has infinite order if and are parallel or ultraparallel, and
(b) has finite order if and only if and meet at an angle which is a rational multiple of .
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1.I.1A
Part IB, 2001 commentDefine uniform continuity for functions defined on a (bounded or unbounded) interval in .
Is it true that a real function defined and uniformly continuous on is necessarily bounded?
Is it true that a real function defined and with a bounded derivative on is necessarily uniformly continuous there?
Which of the following functions are uniformly continuous on :
(i) ;
(ii) ;
(iii) ?
Justify your answers.
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1.II.10A
Part IB, 2001 commentShow that each of the functions below is a metric on the set of functions :
Is the space complete in the metric? Justify your answer.
Show that the set of functions
is a Cauchy sequence with respect to the metric on , yet does not tend to a limit in the metric in this space. Hence, deduce that this space is not complete in the metric.
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2.I.1A
Part IB, 2001 commentState and prove the contraction mapping theorem.
Let , let be the discrete metric on , and let be the metric given by: is symmetric and
Verify that is a metric, and that it is Lipschitz equivalent to .
Define an appropriate function such that is a contraction in the metric, but not in the metric.
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2.II.10A
Part IB, 2001 commentDefine total boundedness for metric spaces.
Prove that a metric space has the Bolzano-Weierstrass property if and only if it is complete and totally bounded.
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3.I.1A
Part IB, 2001 commentDefine what is meant by a norm on a real vector space.
(a) Prove that two norms on a vector space (not necessarily finite-dimensional) give rise to equivalent metrics if and only if they are Lipschitz equivalent.
(b) Prove that if the vector space has an inner product, then for all ,
in the induced norm.
Hence show that the norm on defined by , where , cannot be induced by an inner product.
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3.II.11A
Part IB, 2001 commentProve that if all the partial derivatives of (with ) exist in an open set containing and are continuous at this point, then is differentiable at .
Let
and
At which points of the plane is the partial derivative continuous?
At which points is the function differentiable? Justify your answers.
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4.I.1A
Part IB, 2001 commentLet be a mapping of a metric space into itself such that for all distinct in .
Show that and are continuous functions of .
Now suppose that is compact and let
Show that we cannot have .
[You may assume that a continuous function on a compact metric space is bounded and attains its bounds.]
Deduce that possesses a fixed point, and that it is unique.
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4.II.10A
Part IB, 2001 commentLet be a pointwise convergent sequence of real-valued functions on a closed interval . Prove that, if for every , the sequence is monotonic in , and if all the functions , and are continuous, then uniformly on .
By considering a suitable sequence of functions on , show that if the interval is not closed but all other conditions hold, the conclusion of the theorem may fail.
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1.I.7E
Part IB, 2001 commentState the Cauchy integral formula.
Assuming that the function is analytic in the disc , prove that, for every , it is true that
[Taylor's theorem may be used if clearly stated.]
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1.II.16E
Part IB, 2001 commentLet the function be integrable for all real arguments , such that
and assume that the series
converges uniformly for all .
Prove the Poisson summation formula
where is the Fourier transform of . [Hint: You may show that
or, alternatively, prove that is periodic and express its Fourier expansion coefficients explicitly in terms of .]
Letting , use the Poisson summation formula to evaluate the sum
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Part IB, 2001
commentA complex function is defined for every , where is a non-empty open subset of , and it possesses a derivative at every . Commencing from a formal definition of derivative, deduce the Cauchy-Riemann equations.
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2.II.16E
Part IB, 2001 commentLet be a rational function such that . Assuming that has no real poles, use the residue calculus to evaluate
Given that is an integer, evaluate
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4.I.8F
Part IB, 2001 commentConsider a conformal mapping of the form
where , and . You may assume . Show that any such which maps the unit circle onto itself is necessarily of the form
[Hint: Show that it is always possible to choose .]
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4.II.17F
Part IB, 2001 commentState Jordan's Lemma.
Consider the integral
for real and . The rectangular contour runs from to , to , to and back to , where is infinitesimal and positive. Perform the integral in two ways to show that
for .
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1.I.2H
Part IB, 2001 commentThe even function has the Fourier cosine series
in the interval . Show that
Find the Fourier cosine series of in the same interval, and show that
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1.II.11H
Part IB, 2001 commentUse the substitution to find the general solution of
Find the Green's function , which satisfies
for , subject to the boundary conditions as and as , for each fixed .
Hence, find the solution of the equation
subject to the same boundary conditions.
Verify that both forms of your solution satisfy the appropriate equation and boundary conditions, and match at .
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2.I.2G
Part IB, 2001 commentShow that the symmetric and antisymmetric parts of a second-rank tensor are themselves tensors, and that the decomposition of a tensor into symmetric and antisymmetric parts is unique.
For the tensor having components
find the scalar , vector and symmetric traceless tensor such that
for every vector .
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2.II.11G
Part IB, 2001 commentExplain what is meant by an isotropic tensor.
Show that the fourth-rank tensor
is isotropic for arbitrary scalars and .
Assuming that the most general isotropic tensor of rank 4 has the form , or otherwise, evaluate
where is the position vector and .
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3.I.2G
Part IB, 2001 commentLaplace's equation in the plane is given in terms of plane polar coordinates and in the form
In each of the cases
find the general solution of Laplace's equation which is single-valued and finite.
Solve also Laplace's equation in the annulus with the boundary conditions
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3.II.12H
Part IB, 2001 commentFind the Fourier sine series representation on the interval of the function
The motion of a struck string is governed by the equation
subject to boundary conditions at and for , and to the initial conditions and at .
Obtain the solution for this motion. Evaluate for , and sketch it clearly.
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4.I.2H
Part IB, 2001 commentThe Legendre polynomial satisfies
Show that obeys an equation which can be recast in Sturm-Liouville form and has the eigenvalue . What is the orthogonality relation for for ?
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4.II.11H
Part IB, 2001 commentA curve in the -plane connects the points and has a fixed length . Find an expression for the area of the surface of the revolution obtained by rotating about the -axis.
Show that the area has a stationary value for
where is a constant such that
Show that the latter equation admits a unique positive solution for .
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1.I
Part IB, 2001 commentA quantum mechanical particle of mass and energy encounters a potential step,
Calculate the probability that the particle is reflected in the case .
If is positive, what is the limiting value of when tends to ? If is negative, what is the limiting value of as tends to for fixed ?
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1.II.18F
Part IB, 2001 commentConsider a quantum-mechanical particle of mass moving in a potential well,
(a) Verify that the set of normalised energy eigenfunctions are
and evaluate the corresponding energy eigenvalues .
(b) At time the wavefunction for the particle is only nonzero in the positive half of the well,
Evaluate the expectation value of the energy, first using
and secondly using
where the are the expansion coefficients in
Hence, show that
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2.I
Part IB, 2001 commentConsider a solution of the time-dependent Schrödinger equation for a particle of mass in a potential . The expectation value of an operator is defined as
Show that
where
and that
[You may assume that vanishes as
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2.II.18F
Part IB, 2001 comment(a) Write down the angular momentum operators in terms of and
Verify the commutation relation
Show that this result and its cyclic permutations imply
(b) Consider a wavefunction of the form , where . Show that for a particular value of is an eigenfunction of both and . What are the corresponding eigenvalues?
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3.II.20F
Part IB, 2001 commentA quantum system has a complete set of orthonormalised energy eigenfunctions with corresponding energy eigenvalues
(a) If the time-dependent wavefunction is, at ,
determine for all .
(b) A linear operator acts on the energy eigenfunctions as follows:
Find the eigenvalues of . At time is measured and its lowest eigenvalue is found. At time is measured again. Show that the probability for obtaining the lowest eigenvalue again is
where .
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3.I.10F
Part IB, 2001 commentA particle of rest mass and four-momentum is detected by an observer with four-velocity , satisfying , where the product of two four-vectors and is .
Show that the speed of the detected particle in the observer's rest frame is
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4.I.9F
Part IB, 2001 commentWhat is Einstein's principle of relativity?
Show that a spherical shell expanding at the speed of light, , in one coordinate system , is still spherical in a second coordinate system defined by
where . Why do these equations define a Lorentz transformation?
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4.II.18F
Part IB, 2001 commentA particle of mass is at rest at , in coordinates . At time it decays into two particles and of equal mass . Assume that particle A moves in the negative direction.
(a) Using relativistic energy and momentum conservation compute the energy, momentum and velocity of both particles and
(b) After a proper time , measured in its own rest frame, particle A decays. Show that the spacetime coordinates of this event are
where .
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1.I.6G
Part IB, 2001 commentDetermine the pressure at a depth below the surface of a static fluid of density subject to gravity . A rigid body having volume is fully submerged in such a fluid. Calculate the buoyancy force on the body.
An iceberg of uniform density is observed to float with volume protruding above a large static expanse of seawater of density . What is the total volume of the iceberg?
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1.II.15G
Part IB, 2001 commentA fluid motion has velocity potential given by
where is a constant. Find the corresponding velocity field . Determine .
The time-average of a quantity is defined as .
Show that the time-average of this velocity field at every point is zero.
Write down an expression for the fluid acceleration and find the time-average acceleration at .
Suppose now that . The material particle at at time is marked with dye. Write down equations for its subsequent motion and verify that its position at time is given (correct to terms of order ) as
Deduce the time-average velocity of the dyed particle correct to this order.
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3.I.8G
Part IB, 2001 commentInviscid incompressible fluid occupies the region , which is bounded by a rigid barrier along . At time , a line vortex of strength is placed at position . By considering the flow due to an image vortex at , or otherwise, determine the velocity potential in the fluid.
Derive the position of the original vortex at time .
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3.II.18G
Part IB, 2001 commentState the form of Bernoulli's theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid.
A circular cylinder of radius is immersed in unbounded inviscid fluid of uniform density . The cylinder moves in a prescribed direction perpendicular to its axis, with speed . Use cylindrical polar coordinates, with the direction parallel to the direction of the motion, to find the velocity potential in the fluid.
If depends on time and gravity is negligible, determine the pressure field in the fluid at time . Deduce the fluid force per unit length on the cylinder.
[In cylindrical polar coordinates, .]
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4.I.7G
Part IB, 2001 commentStarting from the Euler equation, derive the vorticity equation for the motion of an inviscid incompressible fluid under a conservative body force, and give a physical interpretation of each term in the equation. Deduce that in a flow field of the form the vorticity of a material particle is conserved.
Find the vorticity for such a flow in terms of the stream function . Deduce that if the flow is steady, there must be a function such that
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4.II.16G
Part IB, 2001 commentA long straight canal has rectangular cross-section with a horizontal bottom and width that varies slowly with distance downstream. Far upstream, has a constant value , the water depth has a constant value , and the velocity has a constant value . Assuming that the water velocity is steady and uniform across the channel, use mass conservation and Bernoulli's theorem, which should be stated carefully, to show that the water depth satisfies
Deduce that for a given value of , a flow of this kind can exist only if is everywhere greater than or equal to a critical value , which is to be determined in terms of .
Suppose that everywhere. At locations where the channel width exceeds , determine graphically, or otherwise, under what circumstances the water depth exceeds
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2.I.5E
Part IB, 2001 commentFind an LU factorization of the matrix
and use it to solve the linear system , where
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2.II.14E
Part IB, 2001 comment(a) Let be an positive-definite, symmetric matrix. Define the Cholesky factorization of and prove that it is unique.
(b) Let be an matrix, , such that . Prove the uniqueness of the "skinny QR factorization"
where the matrix is with orthonormal columns, while is an upper-triangular matrix with positive diagonal elements.
[Hint: Show that you may choose as a matrix that features in the Cholesky factorization of .]
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3.I.6E
Part IB, 2001 commentGiven , let the th-degree polynomial interpolate the values , , where are distinct. Given , find the error in terms of a derivative of .
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3.II.16E
Part IB, 2001 commentLet the monic polynomials , be orthogonal with respect to the weight function , where the degree of each is exactly .
(a) Prove that each , has distinct zeros in the interval .
(b) Suppose that the satisfy the three-term recurrence relation
where . Set
Prove that , and deduce that all the eigenvalues of reside in .
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1.I.3D
Part IB, 2001 commentLet be independent, identically distributed random variables, .
Find a two-dimensional sufficient statistic for , quoting carefully, without proof, any result you use.
What is the maximum likelihood estimator of ?
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1.II.12D
Part IB, 2001 commentWhat is a simple hypothesis? Define the terms size and power for a test of one simple hypothesis against another.
State, without proof, the Neyman-Pearson lemma.
Let be a single random variable, with distribution . Consider testing the null hypothesis is standard normal, , against the alternative hypothesis is double exponential, with density .
Find the test of size , which maximises power, and show that the power is , where and is the distribution function of .
[Hint: if
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2.I.3D
Part IB, 2001 commentSuppose the single random variable has a uniform distribution on the interval and it is required to estimate with the loss function
where .
Find the posterior distribution for and the optimal Bayes point estimate with respect to the prior distribution with density .
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2.II.12D
Part IB, 2001 commentWhat is meant by a generalized likelihood ratio test? Explain in detail how to perform such a test
Let be independent random variables, and let have a Poisson distribution with unknown mean .
Find the form of the generalized likelihood ratio statistic for testing , and show that it may be approximated by
where .
If, for , you found that the value of this statistic was , would you accept ? Justify your answer.
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4.I.3D
Part IB, 2001 commentConsider the linear regression model
, where are given constants, and are independent, identically distributed , with unknown.
Find the least squares estimator of . State, without proof, the distribution of and describe how you would test against , where is given.
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4.II.12D
Part IB, 2001 commentLet be independent, identically distributed random variables, where and are unknown.
Derive the maximum likelihood estimators of , based on . Show that and are independent, and derive their distributions.
Suppose now it is intended to construct a "prediction interval" for a future, independent, random variable . We require
with the probability over the joint distribution of .
Let
By considering the distribution of , find the value of for which
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Part IB, 2001
commentLet be given constants, not all equal.
Use the Lagrangian sufficiency theorem, which you should state clearly, without proof, to minimize subject to the two constraints
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3.II.15D
Part IB, 2001 commentConsider the following linear programming problem,
Formulate the problem in a suitable way for solution by the two-phase simplex method.
Using the two-phase simplex method, show that if then the optimal solution has objective function value , while if the optimal objective function value is .
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4.I.5D
Part IB, 2001 commentExplain what is meant by a two-person zero-sum game with payoff matrix . Write down a set of sufficient conditions for a pair of strategies to be optimal for such a game.
A fair coin is tossed and the result is shown to player I, who must then decide to 'pass' or 'bet'. If he passes, he must pay player II . If he bets, player II, who does not know the result of the coin toss, may either 'fold' or 'call the bet'. If player II folds, she pays player I . If she calls the bet and the toss was a head, she pays player I ; if she calls the bet and the toss was a tail, player I must pay her .
Formulate this as a two-person zero-sum game and find optimal strategies for the two players. Show that the game has value .
[Hint: Player I has four possible moves and player II two.]
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4.II.14D
Part IB, 2001 commentDumbledore Publishers must decide how many copies of the best-selling "History of Hogwarts" to print in the next two months to meet demand. It is known that the demands will be for 40 thousand and 60 thousand copies in the first and second months respectively, and these demands must be met on time. At the beginning of the first month, a supply of 10 thousand copies is available, from existing stock. During each month, Dumbledore can produce up to 40 thousand copies, at a cost of 400 galleons per thousand copies. By having employees work overtime, up to 150 thousand additional copies can be printed each month, at a cost of 450 galleons per thousand copies. At the end of each month, after production and the current month's demand has been satisfied, a holding cost of 20 galleons per thousand copies is incurred.
Formulate a transportation problem, with 5 supply points and 3 demand points, to minimize the sum of production and holding costs during the two month period, and solve it.
[You may assume that copies produced during a month can be used to meet demand in that month.]
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1.I.8B
Part IB, 2001 commentLet be a binary quadratic form with integer coefficients. Define what is meant by the discriminant of , and show that is positive-definite if and only if . Define what it means for the form to be reduced. For any integer , we define the class number to be the number of positive-definite reduced binary quadratic forms (with integer coefficients) with discriminant . Show that is always finite (for negative . Find , and exhibit the corresponding reduced forms.
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1.II.17B
Part IB, 2001 commentLet be a symmetric bilinear form on a finite dimensional vector space over a field of characteristic . Prove that the form may be diagonalized, and interpret the rank of in terms of the resulting diagonal form.
For a symmetric bilinear form on a real vector space of finite dimension , define the signature of , proving that it is well-defined. A subspace of is called null if ; show that has a null subspace of dimension , but no null subspace of higher dimension.
Consider now the quadratic form on given by
Write down the matrix for the corresponding symmetric bilinear form, and calculate . Hence, or otherwise, find the rank and signature of .
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2.I.8B
Part IB, 2001 commentLet be a finite-dimensional vector space over a field . Describe a bijective correspondence between the set of bilinear forms on , and the set of linear maps of to its dual space . If are non-degenerate bilinear forms on , prove that there exists an isomorphism such that for all . If furthermore both are symmetric, show that is self-adjoint (i.e. equals its adjoint) with respect to .
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2.II.17B
Part IB, 2001 commentSuppose is an odd prime and an integer coprime to . Define the Legendre symbol , and state (without proof) Euler's criterion for its calculation.
For any positive integer, we denote by the (unique) integer with and . Let be the number of integers for which is negative. Prove that
Hence determine the odd primes for which 2 is a quadratic residue.
Suppose that are primes congruent to 7 modulo 8 , and let
Show that 2 is a quadratic residue for any prime dividing . Prove that is divisible by some prime . Hence deduce that there are infinitely many primes congruent to 7 modulo 8 .
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3.I.9B
Part IB, 2001 commentLet be the Hermitian matrix
Explaining carefully the method you use, find a diagonal matrix with rational entries, and an invertible (complex) matrix such that , where here denotes the conjugated transpose of .
Explain briefly why we cannot find as above with unitary.
[You may assume that if a monic polynomial with integer coefficients has all its roots rational, then all its roots are in fact integers.]
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3.II.19B
Part IB, 2001 commentLet denote the matrix . Suppose that is a uppertriangular real matrix with strictly positive diagonal entries and that is orthogonal. Verify that .
Prove that any real invertible matrix has a decomposition , where is an orthogonal matrix and is an upper-triangular matrix with strictly positive diagonal entries.
Let now denote a real matrix, and be the decomposition of the previous paragraph. Let denote the matrix with copies of on the diagonal, and zeros elsewhere, and suppose that . Prove that is orthogonal. From this, deduce that the entries of are zero, apart from orthogonal blocks along the diagonal. Show that each has the form , for some upper-triangular matrix with strictly positive diagonal entries. Deduce that and .
[Hint: The invertible matrices with blocks along the diagonal, but with all other entries below the diagonal zero, form a group under matrix multiplication.]
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2.I.4B
Part IB, 2001 commentDefine the terms connected and path connected for a topological space. If a topological space is path connected, prove that it is connected.
Consider the following subsets of :
Let
with the subspace (metric) topology. Prove that is connected.
[You may assume that any interval in (with the usual topology) is connected.]
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2.II.13A
Part IB, 2001 commentState Liouville's Theorem. Prove it by considering
and letting .
Prove that, if is a function analytic on all of with real and imaginary parts and , then either of the conditions:
implies that is constant.
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3.I.3B
Part IB, 2001 commentState a version of Rouché's Theorem. Find the number of solutions (counted with multiplicity) of the equation
inside the open disc , for the cases and 5 .
[Hint: For the case , you may find it helpful to consider the function 2) .]
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3.II.13B
Part IB, 2001 commentIf and are topological spaces, describe the open sets in the product topology on . If the topologies on and are induced from metrics, prove that the same is true for the product.
What does it mean to say that a topological space is compact? If the topologies on and are compact, prove that the same is true for the product.
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4.I.4A
Part IB, 2001 commentLet be analytic in the . Assume the formula
By combining this formula with a complex conjugate version of Cauchy's Theorem, namely
prove that
where is the real part of .
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4.II.13B
Part IB, 2001 commentLet be a punctured disc, and an analytic function on . What does it mean to say that has the origin as (i) a removable singularity, (ii) a pole, and (iii) an essential singularity? State criteria for (i), (ii), (iii) to occur, in terms of the Laurent series for at 0 .
Suppose now that the origin is an essential singularity for . Given any , show that there exists a sequence of points in such that and . [You may assume the fact that an isolated singularity is removable if the function is bounded in some open neighbourhood of the singularity.]
State the Open Mapping Theorem. Prove that if is analytic and injective on , then the origin cannot be an essential singularity. By applying this to the function , or otherwise, deduce that if is an injective analytic function on , then is linear of the form , for some non-zero complex number . [Here, you may assume that injective implies that its derivative is nowhere vanishing.]
Part IB
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A2.7
Part II, 2001 comment(i) Give the definition of the curvature of a plane curve . Show that, if is a simple closed curve, then
(ii) Give the definition of a geodesic on a parametrized surface in . Derive the differential equations characterizing geodesics. Show that a great circle on the unit sphere is a geodesic.
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A3.7
Part II, 2001 comment(i) Give the definition of the surface area of a parametrized surface in and show that it does not depend on the parametrization.
(ii) Let be a differentiable function of . Consider the surface of revolution:
Find a formula for each of the following: (a) The first fundamental form. (b) The unit normal. (c) The second fundamental form. (d) The Gaussian curvature.
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A
Part II, 2001 commentWrite an essay on the Gauss-Bonnet theorem. Make sure that your essay contains a precise statement of the theorem, in its local form, and a discussion of some of its applications, including the global Gauss-Bonnet theorem.
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A1.8
Part II, 2001 comment(i) Show that any graph with minimal degree contains a cycle of length at least . Give examples to show that, for each possible value of , there is a graph with minimal degree but no cycle of length greater than .
(ii) Let be the complete graph with vertices labelled . Prove, from first principles, that there are different spanning trees in . In how many of these spanning trees does the vertex have degree 1 ?
A spanning tree in is chosen at random, with each of the trees being equally likely. Show that the average number of vertices of degree 1 in the random tree is approximately when is large.
Find the average degree of vertices in the random tree.
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A2.8
Part II, 2001 comment(i) Prove that any graph drawn on a compact surface with negative Euler characteristic has a vertex colouring that uses at most
colours.
Briefly discuss whether the result is still true when .
(ii) Prove that a graph is edge-connected if and only if the removal of no set of less than edges from disconnects .
[If you use any form of Menger's theorem, you must prove it.]
Let be a minimal example of a graph that requires colours for a vertex colouring. Show that must be edge-connected.
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A4.9
Part II, 2001 commentWrite an essay on extremal graph theory. Your essay should include proofs of at least two major results and a discussion of variations on these results or their proofs.
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A1.9
Part II, 2001 comment(i) Describe Euclid's algorithm.
Find, in the RSA algorithm, the deciphering key corresponding to the enciphering key 7,527 .
(ii) Explain what is meant by a primitive root modulo an odd prime .
Show that, if is a primitive root modulo , then all primitive roots modulo are given by , where and .
Verify, by Euler's criterion, that 3 is a primitive root modulo 17 . Hence find all primitive roots modulo 17 .
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A3.9
Part II, 2001 comment(i) State the law of quadratic reciprocity.
Prove that 5 is a quadratic residue modulo primes and a quadratic non-residue modulo primes .
Determine whether 5 is a quadratic residue or non-residue modulo 119 and modulo
(ii) Explain what is meant by the continued fraction of a real number . Define the convergents to and write down the recurrence relations satisfied by their numerators and denominators.
Use the continued fraction method to find two solutions in positive integers of the equation .
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A4.10
Part II, 2001 commentAttempt one of the following:
(i) Discuss pseudoprimes and primality testing. Find all bases for which 57 is a Fermat pseudoprime. Determine whether 57 is also an Euler pseudoprime for these bases.
(ii) Write a brief account of various methods for factoring large numbers. Use Fermat factorization to find the factors of 10033. Would Pollard's method also be practical in this instance?
(iii) Show that is divergent, where denotes the -th prime.
Write a brief account of basic properties of the Riemann zeta-function.
State the prime number theorem. Show that it implies that for all sufficiently large positive integers there is a prime satisfying .
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A2.9
Part II, 2001 comment(i) Give brief answers to the following questions.
(a) What is a stream cypher?
(b) Explain briefly why a one-time pad is safe if used only once but becomes unsafe if used many times.
(c) What is a feedback register of length ? What is a linear feedback register of length
(d) A cypher stream is given by a linear feedback register of known length . Show that, given plain text and cyphered text of length , we can find the complete cypher stream.
(e) State and prove a similar result for a general feedback register.
(ii) Describe the construction of a Reed-Muller code. Establish its information rate and its weight
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A2.10
Part II, 2001 comment(i) Let be a directed network with nodes and . Let be a subset of the nodes, be a flow on , and be the divergence of . Describe carefully what is meant by a cut . Define the arc-cut incidence , and the flux of across . Define also the divergence of . Show that .
Now suppose that capacity constraints are specified on each of the arcs. Define the upper cut capacity of . State the feasible distribution problem for a specified divergence , and show that the problem only has a solution if and for all cuts .
(ii) Describe an algorithm to find a feasible distribution given a specified divergence and capacity constraints on each arc. Explain what happens when no feasible distribution exists.
Illustrate the algorithm by either finding a feasible circulation, or demonstrating that one does not exist, in the network given below. Arcs are labelled with capacity constraint intervals.

Part II
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A1.10
Part II, 2001 comment(i) Explain briefly how and why a signature scheme is used. Describe the el Gamal scheme,
(ii) Define a cyclic code. Define the generator of a cyclic code and show that it exists. Prove a necessary and sufficient condition for a polynomial to be the generator of a cyclic code of length .
What is the code? Show that the code associated with , where is a root of in an appropriate field, is Hamming's original code.
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A3.10
Part II, 2001 comment(i) Let be the problem
Explain carefully what it means for the problem to be Strong Lagrangian.
Outline the main steps in a proof that a quadratic programming problem
where is a symmetric positive semi-definite matrix, is Strong Lagrangian.
[You should carefully state the results you need, but should not prove them.]
(ii) Consider the quadratic programming problem:
State necessary and sufficient conditions for to be optimal, and use the activeset algorithm (explaining your steps briefly) to solve the problem starting with initial condition . Demonstrate that the solution you have found is optimal by showing that it satisfies the necessary and sufficient conditions stated previously.
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A4.11
Part II, 2001 commentState the optimal distribution problem. Carefully describe the simplex-on-a-graph algorithm for solving optimal distribution problems when the flow in each arc in the network is constrained to lie in the interval . Explain how the algorithm can be initialised if there is no obvious feasible solution with which to begin. Describe the adjustments that are needed for the algorithm to cope with more general capacity constraints for each arc (where may be finite or infinite).
Part II
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A1.13
Part II, 2001 comment(i) Assume that the -dimensional observation vector may be written as
where is a given matrix of is an unknown vector, and
Let . Find , the least-squares estimator of , and show that
where is a matrix that you should define.
(ii) Show that . Show further for the special case of
where , that
here, is a vector of which every element is one, and , are constants that you should derive.
Hence show that, if is the vector of fitted values, then
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A2.12
Part II, 2001 comment(i) Suppose that are independent random variables, and that has probability density function
Assume that , and that , where is a known 'link' function, are known covariates, and is an unknown vector. Show that
and hence
(ii) The table below shows the number of train miles (in millions) and the number of collisions involving British Rail passenger trains between 1970 and 1984 . Give a detailed interpretation of the output that is shown under this table:
Call:
glm(formula collisions year miles , family poisson)
Coefficients:
(Dispersion parameter for poisson family taken to be 1)
Null deviance: on 13 degrees of freedom
Residual deviance: on 11 degrees of freedom
Number of Fisher Scoring iterations: 4
Part II
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A4.14
Part II, 2001 comment(i) Assume that independent observations are such that
where are given covariates. Discuss carefully how to estimate , and how to test that the model fits.
(ii) Carmichael et al. (1989) collected data on the numbers of 5 -year old children with "dmft", i.e. with 5 or more decayed, missing or filled teeth, classified by social class, and by whether or not their tap water was fluoridated or non-fluoridated. The numbers of such children with dmft, and the total numbers, are given in the table below:
\begin{tabular}{l|ll} Social Class & Fluoridated & Non-fluoridated \ \hline I & & \ II & & \ III & & \ Unclassified & & \end{tabular}
A (slightly edited) version of the output is given below. Explain carefully what model is being fitted, whether it does actually fit, and what the parameter estimates and Std. Errors are telling you. (You may assume that the factors SClass (social class) and Fl (with/without) have been correctly set up.)


Here 'Yes' is the vector of numbers with dmft, taking values , 'Total' is the vector of Total in each category, taking values , and SClass, Fl are the factors corresponding to Social class and Fluoride status, defined in the obvious way.
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A1.14
Part II, 2001 comment(i) A spinless quantum mechanical particle of mass moving in two dimensions is confined to a square box with sides of length . Write down the energy eigenfunctions for the particle and the associated energies.
Show that, for large , the number of states in the energy range is , where
(ii) If, instead, the particle is an electron with magnetic moment moving in an external magnetic field, , show that
Let there be electrons in the box. Explain briefly how to construct the ground state of the system. Let be the Fermi energy. Show that when ,
Show also that the magnetic moment, , of the system in the ground state is
and that the ground state energy is
Part II
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A2.14
Part II, 2001 comment(i) Each particle in a system of identical fermions has a set of energy levels, , with degeneracy , where . Explain why, in thermal equilibrium, the average number of particles with energy is
The physical significance of the parameters and should be made clear.
(ii) A simple model of a crystal consists of a linear array of sites with separation . At the th site an electron may occupy either of two states with probability amplitudes and , respectively. The time-dependent Schrödinger equation governing the amplitudes gives
where .
By examining solutions of the form
show that the energies of the electron fall into two bands given by
Describe briefly how the energy band structure for electrons in real crystalline materials can be used to explain why they are insulators, conductors or semiconductors.
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A4.16
Part II, 2001 commentA harmonic oscillator of frequency is in thermal equilibrium with a heat bath at temperature . Show that the mean number of quanta in the oscillator is
Use this result to show that the density of photons of frequency for cavity radiation at temperature is
By considering this system in thermal equilibrium with a set of distinguishable atoms, derive formulae for the Einstein and coefficients.
Give a brief description of the operation of a laser.
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A1.16
Part II, 2001 comment(i) Introducing the concept of a co-moving distance co-ordinate, explain briefly how the velocity of a galaxy in an isotropic and homogeneous universe is determined by the scale factor . How is the scale factor related to the Hubble constant ?
A homogeneous and isotropic universe has an energy density and a pressure . Use the relation to derive the "fluid equation"
where the overdot indicates differentiation with respect to time, . Given that satisfies the "acceleration equation"
show that the quantity
is time-independent.
The pressure is related to by the "equation of state"
Given that , find for , and hence show that .
(ii) What is meant by the expression "the Hubble time"?
Assuming that and , where is the time now (of the current cosmological era), obtain a formula for the radius of the observable universe.
Given that
for constant , find the values of for which is finite. Given that is finite, show that the age of the universe is less than the Hubble time. Explain briefly, and qualitatively, why this result is to be expected as long as
Part II
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A3.14
Part II, 2001 comment(i) A spherically symmetric star has pressure and mass density , where is distance from the star's centre. Stating without proof any theorems you may need, show that mechanical equilibrium implies the Newtonian pressure support equation
where is the mass within radius and .
Write down an integral expression for the total gravitational potential energy, . Use this to derive the "virial theorem"
when is the average pressure.
(ii) Given that the total kinetic energy, , of a spherically symmetric star is related to its average pressure by the formula
for constant , use the virial theorem (stated in part (i)) to determine the condition on needed for gravitational binding. State the relation between pressure and "internal energy" for an ideal gas of non-relativistic particles. What is the corresponding relation for ultra-relativistic particles? Hence show that the formula applies in these cases, and determine the values of .
Why does your result imply a maximum mass for any star, whatever the source of its pressure? What is the maximum mass, approximately, for stars supported by
(a) thermal pressure,
(b) electron degeneracy pressure (White Dwarf),
(c) neutron degeneracy pressure (Neutron Star).
A White Dwarf can accrete matter from a companion star until its mass exceeds the Chandrasekar limit. Explain briefly the process by which it then evolves into a neutron star.
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A4.18
Part II, 2001 comment(i) Given that is the number of eigenstates of a gas particle with momentum between and , write down the Bose-Einstein distribution for the average number of particles with momentum between and , as a function of temperature and chemical potential .
Given that and for a gas of photons, obtain a formula for the energy density at temperature in the form
where is a function of the photon frequency that you should determine. Hence show that the value of at the maximum of is proportional to .
A thermally isolated photon gas undergoes a slow change of its volume . Why is unaffected by this change? Use this fact to show that remains constant.
(ii) According to the "Hot Big Bang" theory, the Universe evolved by expansion from an earlier state in which it was filled with a gas of electrons, protons and photons (with ) at thermal equilibrium at a temperature such that
where is the electron mass and is the binding energy of a hydrogen atom. Why must the composition have been different when ? Why must it change as the temperature falls to ? Why does this lead to a thermal decoupling of radiation from matter?
The baryon number of the Universe can be taken to be the number of protons, either as free particles or as hydrogen atom nuclei. Let be the baryon number density and the photon number density. Why is the ratio unchanged by the expansion of the universe? Given that , obtain an estimate of the temperature at which decoupling occurs, as a function of and . How does this decoupling lead to the concept of a "surface of last scattering" and a prediction of a Cosmic Microwave Background Radiation (CMBR)?
Part II
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A1.17
Part II, 2001 comment(i) Let be a surjective homomorphism between two groups, and . If is a representation of , show that for is a representation of and, if is irreducible, show that is also irreducible. Show further that is a representation of , where for and (with ). Deduce that the characters of , respectively, satisfy
(ii) is the symmetry group of rotations and reflections of a square. If is a rotation by about the centre of the square and is a reflection in one of its symmetry axes, then . Given that the conjugacy classes are and derive the character table of .
Let be the Hamiltonian of a particle moving in a central potential. The symmetry ensures that the energy eigenvalues of are the same for all the angular momentum eigenstates in a given irreducible representation. Therefore, the energy eigenvalues of are labelled with and . Assume now that in a crystal the symmetry is reduced to a symmetry by an additional term of the total Hamiltonian, . Find how the eigenstates in the irreducible representation with (the D-wave orbital) decompose into irreducible representations of . You may assume that the character, , of a group element of , in a representation labelled by is given by
where is a rotation angle and is the eigenvalue of the total angular momentum, .
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A3.15
Part II, 2001 comment(i) The pions form an isospin triplet with and , whilst the nucleons form an isospin doublet with and . Consider the isospin representation of two-particle states spanned by the basis
State which irreducible representations are contained in this representation and explain why is an isospin eigenstate.
Using
where is the isospin ladder operator, write the isospin eigenstates in terms of the basis, .
(ii) The Lie algebra of generators of is spanned by the operators satisfying the commutator algebra and . Let be an eigenvector of such that . The vector space together with the action of an arbitrary su(2) operator on defined by
forms a representation (not necessarily reducible) of . Show that if is nontrivial then it is an eigenvector of and find its eigenvalue. Given that show that and satisfy
By solving these equations evaluate . Show that . Hence show that is contained in a proper sub-representation of .
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A1.18
Part II, 2001 comment(i) The diffusion equation for a spherically-symmetric concentration field is
where is the radial coordinate. Find and sketch the similarity solution to (1) which satisfies as and constant, assuming it to be of the form
where and are numbers to be found.
(ii) A two-dimensional piece of heat-conducting material occupies the region (in plane polar coordinates). The surfaces are maintained at a constant temperature ; at the surface the boundary condition on temperature is
where is a constant. Show that the temperature, which satisfies the steady heat conduction equation
is given by a Fourier series of the form
where are to be found.
In the limits and , show that
given that
Explain how, in these limits, you could have obtained this result much more simply.
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A3.16
Part II, 2001 comment(i) Incompressible fluid of kinematic viscosity occupies a parallel-sided channel . The wall is moving parallel to itself, in the direction, with velocity , where is time and are real constants. The fluid velocity satisfies the equation
write down the boundary conditions satisfied by .
Assuming that
where , find the complex constants . Calculate the velocity (in real, not complex, form) in the limit .
(ii) Incompressible fluid of viscosity fills the narrow gap between the rigid plane , which moves parallel to itself in the -direction with constant speed , and the rigid wavy wall , which is at rest. The length-scale, , over which varies is much larger than a typical value, , of .
Assume that inertia is negligible, and therefore that the governing equations for the velocity field and the pressure are
Use scaling arguments to show that these equations reduce approximately to
Hence calculate the velocity , the flow rate
and the viscous shear stress exerted by the fluid on the plane wall,
in terms of and .
Now assume that , where and , and that is periodic in with wavelength . Show that
and calculate correct to . Does increasing the amplitude of the corrugation cause an increase or a decrease in the force required to move the plane at the chosen speed
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A4.19
Part II, 2001 commentFluid flows in the -direction past the infinite plane with uniform but timedependent velocity , where is a positive function with timescale . A long region of the plane, , is heated and has temperature , where are constants ; the remainder of the plane is insulating . The fluid temperature far from the heated region is . A thermal boundary layer is formed over the heated region. The full advection-diffusion equation for temperature is
where is the thermal diffusivity. By considering the steady case , derive a scale for the thickness of the boundary layer, and explain why the term in (1) can be neglected if .
Neglecting , use the change of variables
to transform the governing equation to
Write down the boundary conditions to be satisfied by in the region .
In the case in which is slowly-varying, so , consider a solution for in the form
Explain why is independent of and .
Henceforth take . Calculate and show that
where satisfies the ordinary differential equation
State the boundary conditions on .
The heat transfer per unit length of the heated region is . Use the above results to show that the total rate of heat transfer is
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A1.19
Part II, 2001 comment(i) From the surface of a flat Earth, an explosive source emits P-waves downward into a horizontal homogeneous elastic layer of uniform thickness and P-wave speed overlying a lower layer of infinite depth and P-wave speed , where . A line of seismometers on the surface records the travel time as a function of distance from the source for the various arrivals along different ray paths.
Sketch the ray paths associated with the direct, reflected and head waves arriving at a given position. Calculate the travel times of the direct and reflected waves, and sketch the corresponding travel-time curves. Hence explain how to estimate and from the recorded arrival times. Explain briefly why head waves are only observed beyond a minimum distance from the source and why they have a travel-time curve of the form for .
[You need not calculate or .]
(ii) A plane -wave in a homogeneous elastic solid has displacement proportional to . Express the slowness vector in terms of the wavevector and . Deduce an equation for in terms of and the S-wave speed .
A homogeneous elastic layer of uniform thickness , S-wave speed and shear modulus has a stress-free surface and overlies a lower layer of infinite depth, S-wave speed and shear modulus . Find the vertical structure of Love waves with displacement proportional to , and show that the horizontal phase speed obeys
By sketching both sides of the equation as a function of in show that at least one mode exists for every value of .
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A2.16
Part II, 2001 comment(i) In a reference frame rotating with constant angular velocity the equations of motion for an inviscid, incompressible fluid of density in a gravitational field are
Define the Rossby number and explain what is meant by geostrophic flow.
Derive the vorticity equation
Recall that .]
Give a physical interpretation for the term .
(ii) Consider the rotating fluid of part (i), but now let be constant and absorb the effects of gravity into a modified pressure . State the linearized equations of motion and the linearized vorticity equation for small-amplitude motions (inertial waves).
Use the linearized equations of motion to show that
Calculate the time derivative of the curl of the linearized vorticity equation. Hence show that
Deduce the dispersion relation for waves proportional to . Show that . Show further that if then .
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A4.20
Part II, 2001 commentWrite down expressions for the phase speed and group velocity in one dimension for general waves of the form with dispersion relation . Briefly indicate the physical significance of and for a wavetrain of finite size.
The dispersion relation for internal gravity waves with wavenumber in an incompressible stratified fluid with constant buoyancy frequency is
Calculate the group velocity and show that it is perpendicular to . Show further that the horizontal components of and have the same sign and that the vertical components have the opposite sign.
The vertical velocity of small-amplitude internal gravity waves is governed by
where is the horizontal part of the Laplacian and is constant.
Find separable solutions to of the form corresponding to waves with constant horizontal phase speed . Comment on the nature of these solutions for and for .
A semi-infinite stratified fluid occupies the region above a moving lower boundary . Construct the solution to for the case , where and are constants and .
Sketch the orientation of the wavecrests, the propagation direction and the group velocity for the case .
Part II
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A2.17
Part II, 2001 comment(i) A certain physical quantity can be represented by the series in , but the series diverges for . Describe the Euler transformation to a new series which may enable to be computed for . Give the first four terms of the new series.
Describe briefly the disadvantages of the method.
(ii) The series has partial sums . Describe Shanks' method to approximate by
giving expressions for and .
Denote by and the values of and respectively derived from these expressions using and for some fixed . Now let be the value of obtained from with . Show that, if ,
If, in fact, the partial sums satisfy
with , show that
where is to be found.
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A3.17
Part II, 2001 comment(i) The function satisfies the differential equation
where and are constants, with boundary conditions . By integrating this equation or otherwise, show that must also satisfy the integral equation
and find the functions and .
(ii) Solve the integral equation
by finding an ordinary differential equation satisfied by together with boundary conditions.
Now solve the integral equation by the method of successive approximations and show that the solutions are the same.
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A4.21
Part II, 2001 commentThe equation
where is a real square matrix and a column vector, has a simple eigenvalue with corresponding right-eigenvector . Show how to find expressions for the perturbed eigenvalue and right-eigenvector solutions of
to first order in , where is a vector function of . State clearly any assumptions you make.
If is and has a complete set of right-eigenvectors , which span and correspond to separate eigenvalues , find an expression for the first-order perturbation to in terms of the and the corresponding lefteigenvectors of .
Find the normalised eigenfunctions and eigenvalues of the equation
with . Let these be the zeroth order approximations to the eigenfunctions of
with and where is a function of . Show that the first-order perturbations of the eigenvalues are given by
Part II
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A2.18
Part II, 2001 comment(i) Establish two conservation laws for the equation
State sufficient boundary conditions that should satisfy for the conservation laws to be valid.
(ii) The equation
models traffic flow on a single-lane road, where represents the density of cars, and is a given function of . By considering the rate of change of the integral
show that represents the velocity of the cars.
Suppose now that (in suitable units), and that everywhere. Assume that a queue is building up at a traffic light at , so that, when the light turns green at ,
For this problem, find and sketch the characteristics in the plane, for , paying particular attention to those emerging from the point . Show that a shock forms at . Find the density of cars for , and all .
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A3.18
Part II, 2001 comment(i) The so-called breather solution of the sine-Gordon equation is
Describe qualitatively the behaviour of , for , when , when , and when . Explain how this solution can be interpreted in terms of motion of a kink and an antikink. Estimate the greatest separation of the kink and antikink.
(ii) The field obeys the nonlinear wave equation
where the potential has the form
Show that and are stable constant solutions.
Find a steady wave solution satisfying the boundary conditions as as . What constraint is there on the velocity
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A1.1 B1.1
Part II, 2001 comment(i) Let be an irreducible Markov chain on the finite state space with transition matrix and invariant distribution . What does it mean to say that is reversible in equilibrium?
Show that is reversible in equilibrium if and only if for all .
(ii) A finite connected graph has vertex set and edge set , and has neither loops nor multiple edges. A particle performs a random walk on , moving at each step to a randomly chosen neighbour of the current position, each such neighbour being picked with equal probability, independently of all previous moves. Show that the unique invariant distribution is given by where is the degree of vertex .
A rook performs a random walk on a chessboard; at each step, it is equally likely to make any of the moves which are legal for a rook. What is the mean recurrence time of a corner square. (You should give a clear statement of any general theorem used.)
[A chessboard is an square grid. A legal move is one of any length parallel to the axes.]
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A2.1
Part II, 2001 comment(i) The fire alarm in Mill Lane is set off at random times. The probability of an alarm during the time-interval is where the 'intensity function' may vary with the time . Let be the number of alarms by time , and set . Show, subject to reasonable extra assumptions to be stated clearly, that satisfies
Deduce that has the Poisson distribution with parameter .
(ii) The fire alarm in Clarkson Road is different. The number of alarms by time is such that
where , and . Show, subject to suitable extra conditions, that satisfies a set of differential-difference equations to be specified. Deduce without solving these equations in their entirety that has mean , and find the variance of .
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A3.1 B3.1
Part II, 2001 comment(i) Explain what is meant by the transition semigroup of a Markov chain in continuous time. If the state space is finite, show under assumptions to be stated clearly, that for some matrix . Show that a distribution satisfies if and only if for all , and explain the importance of such .
(ii) Let be a continuous-time Markov chain on the state space with generator
Show that the transition semigroup is given by
where .
For , let
For a continuous-time chain , let be a matrix with entry
, for . Show that there is a chain with if and only if .
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A4.1
Part II, 2001 commentWrite an essay on the convergence to equilibrium of a discrete-time Markov chain on a countable state-space. You should include a discussion of the existence of invariant distributions, and of the limit theorem in the non-null recurrent case.
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A1.2 B1.2
Part II, 2001 comment(i) Show that Newton's equations in Cartesian coordinates, for a system of particles at positions , in a potential , imply Lagrange's equations in a generalised coordinate system
that is,
where being the total kinetic energy and the total potential energy.
(ii) Consider a light rod of length , free to rotate in a vertical plane (the plane), but with one end forced to move in the -direction. The other end of the rod is attached to a heavy mass upon which gravity acts in the negative direction.
(a) Write down the Lagrangian for the system.
(b) Show that, if is stationary, the rod has two equilibrium positions, one stable and the other unstable.
(c) The end at is now forced to move with constant acceleration, . Show that, once more, there is one stable equilibrium value of the angle the rod makes with the vertical, and find it.
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A2.2 B2.1
Part II, 2001 comment(i) An axially symmetric top rotates freely about a fixed point on its axis. The principal moments of inertia are and the centre of gravity is a distance from
Define the three Euler angles and , specifying the orientation of the top. Use Lagrange's equations to show that there are three conserved quantities in the motion. Interpret them physically.
(ii) Initially the top is spinning with angular speed about , with vertical, before it is slightly disturbed.
Show that, in the subsequent motion, stays close to zero if , but if this condition fails then attains a maximum value given approximately by
Why is this only an approximation?
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A3.2
Part II, 2001 comment(i) (a) Write down Hamilton's equations for a dynamical system. Under what condition is the Hamiltonian a constant of the motion? What is the condition for one of the momenta to be a constant of the motion?
(b) Explain the notion of an adiabatic invariant. Give an expression, in terms of Hamiltonian variables, for one such invariant.
(ii) A mass is attached to one end of a straight spring with potential energy , where is a constant and is the length. The other end is fixed at a point . Neglecting gravity, consider a general motion of the mass in a plane containing . Show that the Hamiltonian is given by
where is the angle made by the spring relative to a fixed direction, and are the generalised momenta. Show that and the energy are constants of the motion, using Hamilton's equations.
If the mass moves in a non-circular orbit, and the spring constant is slowly varied, the orbit gradually changes. Write down the appropriate adiabatic invariant . Show that is proportional to
where
Consider an orbit for which is zero. Show that, as is slowly varied, the energy , for a constant which should be found.
[You may assume without proof that
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A4.2
Part II, 2001 comment(i) Consider a particle of charge and mass , moving in a stationary magnetic field B. Show that Lagrange's equations applied to the Lagrangian
where is the vector potential such that , lead to the correct Lorentz force law. Compute the canonical momentum , and show that the Hamiltonian is .
(ii) Expressing the velocity components in terms of the canonical momenta and co-ordinates for the above system, derive the following formulae for Poisson brackets: (b) ; (c) ; (d) .
(a) , for any functions ;
Now consider a particle moving in the field of a magnetic monopole,
Show that , where . Explain why this means that is conserved.
Show that, if , conservation of implies that the particle moves in a plane perpendicular to . What type of surface does the particle move on if ?
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A1
Part II, 2001 comment(i) Define the adjoint of a bounded, linear map on the Hilbert space . Find the adjoint of the map
where and is the linear map .
Now let be an incomplete inner product space and a bounded, linear map. Is it always true that there is an adjoint ?
(ii) Let be the space of analytic functions on the unit disc for which
You may assume that this is a Hilbert space for the inner product:
Show that the functions form an orthonormal sequence in when the constants are chosen appropriately.
Prove carefully that every function can be written as the sum of a convergent series in with .
For each smooth curve in the disc starting from 0 , prove that
is a continuous, linear map. Show that the norm of satisfies
where is the endpoint of .
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A2.3 B2.2
Part II, 2001 comment(i) State the Stone-Weierstrass theorem for complex-valued functions. Use it to show that the trigonometric polynomials are dense in the space of continuous, complexvalued functions on the unit circle with the uniform norm.
Show further that, for , the th Fourier coefficient
tends to 0 as tends to infinity.
(ii) (a) Let be a normed space with the property that the series converges whenever is a sequence in with convergent. Show that is a Banach space.
(b) Let be a compact metric space and a closed subset of . Let be the map sending to its restriction to . Show that is a bounded, linear map and that its image is a subalgebra of separating the points of
Show further that, for each function in the image of , there is a function with and . Deduce that every continuous, complexvalued function on can be extended to a continuous function on all of .
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A3.3 B3.2
Part II, 2001 comment(i) Define the notion of a measurable function between measurable spaces. Show that a continuous function is measurable with respect to the Borel -fields on and .
By using this, or otherwise, show that, when are measurable with respect to some -field on and the Borel -field on , then is also measurable.
(ii) State the Monotone Convergence Theorem for -valued functions. Prove the Dominated Convergence Theorem.
[You may assume the Monotone Convergence Theorem but any other results about integration that you use will need to be stated carefully and proved.]
Let be the real Banach space of continuous real-valued functions on with the uniform norm. Fix and define
Show that is a bounded, linear map with norm
Is it true, for every choice of , that there is function with and ?
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A4.3
Part II, 2001 commentWrite an account of the classical sequence spaces: and . You should define them, prove that they are Banach spaces, and discuss their properties, including their dual spaces. Show that is inseparable but that and for are separable.
Prove that, if is an isomorphism between two Banach spaces, then
is an isomorphism between their duals.
Hence, or otherwise, show that no two of the spaces are isomorphic.
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A1.4 B1.3
Part II, 2001 comment(i) Define the notion of a Sylow -subgroup of a finite group , and state a theorem concerning the number of them and the relation between them.
(ii) Show that any group of order 30 has a non-trivial normal subgroup. Is it true that every group of order 30 is commutative?
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A2.4 B2.3
Part II, 2001 comment(i) Show that the ring is a field. How many elements does it have?
(ii) Let be as in (i). By considering what happens to a chosen basis of the vector space , or otherwise, find the order of the groups and .
By considering the set of lines in , or otherwise, show that is a subgroup of the symmetric group , and identify this subgroup.
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A3.4
Part II, 2001 comment(i) Let be the cyclic subgroup of generated by the matrix , acting on the polynomial ring . Determine the ring of invariants .
(ii) Determine when is the cyclic group generated by .
[Hint: consider the eigenvectors.]
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A4.4
Part II, 2001 commentShow that the ring is Euclidean, where .
Show that a prime number is reducible in if and only if .
Which prime numbers can be written in the form with (and why)?
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A B
Part II, 2001 comment(i) Write down the two Maxwell equations that govern steady magnetic fields. Show that the boundary conditions satisfied by the magnetic field on either side of a current sheet, , with unit normal to the sheet , are
State without proof the force per unit area on .
(ii) Conducting gas occupies the infinite slab . It carries a steady current and a magnetic field where , depend only on . The pressure is . The equation of hydrostatic equilibrium is . Write down the equations to be solved in this case. Show that is independent of . Using the suffixes 1,2 to denote values at , respectively, verify that your results are in agreement with those of Part (i) in the case of .
Suppose that
Find everywhere in the slab.
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A2.5
Part II, 2001 comment(i) Write down the expression for the electrostatic potential due to a distribution of charge contained in a volume . Perform the multipole expansion of taken only as far as the dipole term.
(ii) If the volume is the sphere and the charge distribution is given by
where are spherical polar coordinates, calculate the charge and dipole moment. Hence deduce as far as the dipole term.
Obtain an exact solution for by solving the boundary value problem using trial solutions of the forms
and
Show that the solution obtained from the multipole expansion is in fact exact for .
[You may use without proof the result
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A3.5 B3.3
Part II, 2001 comment(i) Develop the theory of electromagnetic waves starting from Maxwell equations in vacuum. You should relate the wave-speed to and and establish the existence of plane, plane-polarized waves in which takes the form
You should give the form of the magnetic field in this case.
(ii) Starting from Maxwell's equation, establish Poynting's theorem.
where and . Give physical interpretations of and the theorem.
Compute the averages over space and time of and for the plane wave described in (i) and relate them. Comment on the result.
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A4.5
Part II, 2001 commentWrite down the form of Ohm's Law that applies to a conductor if at a point it is moving with velocity .
Use two of Maxwell's equations to prove that
where is a moving closed loop, is the velocity at the point on , and is a surface spanning . The time derivative on the right hand side accounts for changes in both and B. Explain briefly the physical importance of this result.
Find and sketch the magnetic field described in the vector potential
in cylindrical polar coordinates , where is constant.
A conducting circular loop of radius and resistance lies in the plane with its centre on the -axis.
Find the magnitude and direction of the current induced in the loop as changes with time, neglecting self-inductance.
At time the loop is at rest at . For time the loop moves with constant velocity . Ignoring the inertia of the loop, use energy considerations to find the force necessary to maintain this motion.
[ In cylindrical polar coordinates
Part II
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A1.6
Part II, 2001 comment(i) Given a differential equation for , explain what it means to say that the solution is given by a flow . Define the orbit, , through a point and the -limit set, , of . Define also a homoclinic orbit to a fixed point . Sketch a flow in with a homoclinic orbit, and identify (without detailed justification) the -limit sets for each point in your diagram.
(ii) Consider the differential equations
Transform the equations to polar coordinates in the plane. Solve the equation for to find , and hence find . Hence, or otherwise, determine (with justification) the -limit set for all points .
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A2.6 B2.4
Part II, 2001 comment(i) Define a Liapounov function for a flow on . Explain what it means for a fixed point of the flow to be Liapounov stable. State and prove Liapounov's first stability theorem.
(ii) Consider the damped pendulum
where . Show that there are just two fixed points (considering the phase space as an infinite cylinder), and that one of these is the origin and is Liapounov stable. Show further that the origin is asymptotically stable, and that the the -limit set of each point in the phase space is one or other of the two fixed points, justifying your answer carefully.
[You should state carefully any theorems you use in your answer, but you need not prove them.]
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A3.6 B3.4
Part II, 2001 comment(i) Define a hyperbolic fixed point of a flow determined by a differential equation where and is (i.e. differentiable). State the Hartman-Grobman Theorem for flow near a hyperbolic fixed point. For nonlinear flows in with a hyperbolic fixed point , does the theorem necessarily allow us to distinguish, on the basis of the linearized flow near between (a) a stable focus and a stable node; and (b) a saddle and a stable node? Justify your answers briefly.
(ii) Show that the system:
has a fixed point on the -axis. Show that there is a bifurcation at and determine the stability of the fixed point for and for .
Make a linear change of variables of the form , where and are constants to be determined, to bring the system into the form:
and hence determine whether the periodic orbit produced in the bifurcation is stable or unstable, and whether it exists in or .
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A4.6
Part II, 2001 commentWrite a short essay about periodic orbits in flows in two dimensions. Your essay should include criteria for the existence and non-existence of periodic orbits, and should mention (with sketches) at least two bifurcations that create or destroy periodic orbits in flows as a parameter is altered (though a detailed analysis of any bifurcation is not required).
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A B1.12
Part II, 2001 comment(i) What is the Halting Problem? What is an unsolvable problem?
(ii) Prove that the Halting Problem is unsolvable. Is it decidable whether or not a machine halts with input zero?
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B2.11
Part II, 2001 commentLet be an arbitrary set, and the power set of . For a subset of , the dual of is the set .
(i) Show that .
Show that for a family of subsets of
(ii) Consider . Show that , is a chain-complete poset.
State Zorn's lemma and use it to deduce that there exists with .
Show that if then the following hold:
is closed under superset; for all contains either or .
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A3.8 B3.11
Part II, 2001 comment(i) Write down a set of axioms for the theory of dense linear order with a bottom element but no top element.
(ii) Prove that this theory has, up to isomorphism, precisely one countable model.
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A4.8 B4.10
Part II, 2001 commentWhat is a wellfounded relation, and how does wellfoundedness underpin wellfounded induction?
A formula with two free variables defines an -automorphism if for all there is a unique , the function , defined by if and only if , is a permutation of the universe, and we have .
Use wellfounded induction over to prove that all formulæ defining -automorphisms are equivalent to .
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A1.12 B1.15
Part II, 2001 comment(i) What are the main approaches by which prior distributions are specified in Bayesian inference?
Define the risk function of a decision rule . Given a prior distribution, define what is meant by a Bayes decision rule and explain how this is obtained from the posterior distribution.
(ii) Dashing late into King's Cross, I discover that Harry must have already boarded the Hogwarts Express. I must therefore make my own way onto platform nine and threequarters. Unusually, there are two guards on duty, and I will ask one of them for directions. It is safe to assume that one guard is a Wizard, who will certainly be able to direct me, and the other a Muggle, who will certainly not. But which is which? Before choosing one of them to ask for directions to platform nine and three-quarters, I have just enough time to ask one of them "Are you a Wizard?", and on the basis of their answer I must make my choice of which guard to ask for directions. I know that a Wizard will answer this question truthfully, but that a Muggle will, with probability , answer it untruthfully.
Failure to catch the Hogwarts Express results in a loss which I measure as 1000 galleons, there being no loss associated with catching up with Harry on the train.
Write down an exhaustive set of non-randomised decision rules for my problem and, by drawing the associated risk set, determine my minimax decision rule.
My prior probability is that the guard I ask "Are you a Wizard?" is indeed a Wizard. What is my Bayes decision rule?
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A2.11 B2.16
Part II, 2001 comment(i) Let be independent, identically-distributed random variables, .
Find a minimal sufficient statistic for .
Let and . Write down the distribution of , and hence show that is ancillary. Explain briefly why the Conditionality Principle would lead to inference about being drawn from the conditional distribution of given .
What is the maximum likelihood estimator of ?
(ii) Describe briefly the Bayesian approach to predictive inference,
Let be independent, identically-distributed random variables, with both unknown. Derive the maximum likelihood estimators of based on , and state, without proof, their joint distribution.
Suppose that it is required to construct a prediction interval
for a future, independent, random variable with the same distribution, such that
with the probability over the joint distribution of . Let
where , and , with the distribution function of .
Show that .
By considering the distribution of , or otherwise, show that
and show how to construct an interval with
[Hint: if has the -distribution with degrees of freedom and is defined by then for .]
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A3.12 B3.15
Part II, 2001 comment(i) Explain what is meant by a uniformly most powerful unbiased test of a null hypothesis against an alternative.
Let be independent, identically distributed random variables, with known. Explain how to construct a uniformly most powerful unbiased size test of the null hypothesis that against the alternative that .
(ii) Outline briefly the Bayesian approach to hypothesis testing based on Bayes factors.
Let the distribution of be as in (i) above, and suppose we wish to test, as in (i), against the alternative . Suppose we assume a prior for under the alternative. Find the form of the Bayes factor , and show that, for fixed as .
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A4.13 B4.15
Part II, 2001 commentWrite an account, with appropriate examples, of one of the following:
(a) Inference in multi-parameter exponential families;
(b) Asymptotic properties of maximum-likelihood estimators and their use in hypothesis testing;
(c) Bootstrap inference.
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A1.11 B1.16
Part II, 2001 comment(i) The price of the stock in the binomial model at time , is , where are independent, identically-distributed random variables with and the initial price is a constant. Denote the fixed interest rate on the bank account by , where , and let the discount factor . Determine the unique value for which the sequence is a martingale.
Explain briefly the significance of for the pricing of contingent claims in the model.
(ii) Let denote the first time that a standard Brownian motion reaches the level . Prove that for ,
where is the standard normal distribution function.
Suppose that and represent the prices at time of two different stocks with initial prices 1 and 2 , respectively; the prices evolve so that they may be represented as and , respectively, where and are independent standard Brownian motions and and are constants. Let denote the first time, if ever, that the prices of the two stocks are the same. Determine , for .
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A3.11 B3.16
Part II, 2001 comment(i) Suppose that is a random variable having the normal distribution with and .
For positive constants show that
where is the standard normal distribution function.
In the context of the Black-Scholes model, derive the formula for the price at time 0 of a European call option on the stock at strike price and maturity time when the interest rate is and the volatility of the stock is .
(ii) Let denote the price of the call option in the Black-Scholes model specified in (i). Show that and sketch carefully the dependence of on the volatility (when the other parameters in the model are held fixed).
Now suppose that it is observed that the interest rate lies in the range and when it changes it is linked to the volatility by the formula . Consider a call option at strike price , where is the stock price at time 0 . There is a small increase in the interest rate; will the price of the option increase or decrease (assuming that the stock price is unaffected)? Justify your answer carefully.
[You may assume that the function is decreasing in , and increases to as , where is the standard-normal distribution function and .]
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A4.12 B4.16
Part II, 2001 commentWrite an essay on the mean-variance approach to portfolio selection in a one-period model. Your essay should contrast the solution in the case when all the assets are risky with that for the case when there is a riskless asset.
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A2.13 B2.21
Part II, 2001 comment(i) Hermitian operators , satisfy . The eigenvectors , satisfy and . By differentiating with respect to verify that
and hence show that
Show that
and
(ii) A quantum system has Hamiltonian , where is a small perturbation. The eigenvalues of are . Give (without derivation) the formulae for the first order and second order perturbations in the energy level of a non-degenerate state. Suppose that the th energy level of has degenerate states. Explain how to determine the eigenvalues of corresponding to these states to first order in .
In a particular quantum system an orthonormal basis of states is given by , where are integers. The Hamiltonian is given by
where and unless and are both even.
Obtain an expression for the ground state energy to second order in the perturbation, . Find the energy eigenvalues of the first excited state to first order in the perturbation. Determine a matrix (which depends on two independent parameters) whose eigenvalues give the first order energy shift of the second excited state.
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A3.13 B3.21
Part II, 2001 comment(i) Write the Hamiltonian for the harmonic oscillator,
in terms of creation and annihilation operators, defined by
Obtain an expression for by using the usual commutation relation between and . Deduce the quantized energy levels for this system.
(ii) Define the number operator, , in terms of creation and annihilation operators, and . The normalized eigenvector of with eigenvalue is . Show that .
Determine and in the basis defined by .
Show that
Verify the relation
by considering the action of both sides of the equation on an arbitrary basis vector.
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A4.15 B4.22
Part II, 2001 comment(i) The two states of a spin- particle corresponding to spin pointing along the axis are denoted by and . Explain why the states
correspond to the spins being aligned along a direction at an angle to the direction.
The spin- 0 state of two spin- particles is
Show that this is independent of the direction chosen to define . If the spin of particle 1 along some direction is measured to be show that the spin of particle 2 along the same direction is determined, giving its value.
[The Pauli matrices are given by
(ii) Starting from the commutation relation for angular momentum in the form
obtain the possible values of , where are the eigenvalues of and are the eigenvalues of . Show that the corresponding normalized eigenvectors, , satisfy
and that
The state is defined by
for any complex . By expanding the exponential show that . Verify that
and hence show that
If verify that is a solution of the time-dependent Schrödinger equation.
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A1.15 B1.24
Part II, 2001 comment(i) The metric of any two-dimensional curved space, rotationally symmetric about a point , can by suitable choice of coordinates be written locally in the form
where at away from , and . Labelling the coordinates as , show that the Christoffel symbols and are each zero, and compute the non-zero Christoffel symbols and .
The Ricci tensor is defined by
where a comma denotes a partial derivative. Show that and that
(ii) Suppose further that, in a neighbourhood of , the Ricci scalar takes the constant value . Find a second order differential equation, which you should denote by , for .
This space of constant Ricci scalar can, by a suitable coordinate transformation , leaving invariant, be written locally as
By studying this coordinate transformation, or otherwise, find and in terms of (up to a constant of integration). Deduce that
where is a positive constant and verify that your equation for holds.
[Note that
Part II
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A2.15 B2.23
Part II, 2001 comment(i) Show that the geodesic equation follows from a variational principle with Lagrangian
where the path of the particle is , and is an affine parameter along that path.
(ii) The Schwarzschild metric is given by
Consider a photon which moves within the equatorial plane . Using the above Lagrangian, or otherwise, show that
for constants and . Deduce that
Assume further that the photon approaches from infinity. Show that the impact parameter is given by
By considering the equation , or otherwise
(a) show that, if , the photon is deflected but not captured by the black hole;
(b) show that, if , the photon is captured;
(c) describe, with justification, the qualitative form of the photon's orbit in the case .
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A4.17 B4.25
Part II, 2001 commentDiscuss how Einstein's theory of gravitation reduces to Newton's in the limit of weak fields. Your answer should include discussion of: (a) the field equations; (b) the motion of a point particle; (c) the motion of a pressureless fluid.
[The metric in a weak gravitational field, with Newtonian potential , may be taken as
The Riemann tensor is
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A1.20 B1.20
Part II, 2001 comment(i) Let be a symmetric matrix such that
Prove that it is positive definite.
(ii) Prove that both Jacobi and Gauss-Seidel methods for the solution of the linear system , where the matrix obeys the conditions of (i), converge.
[You may quote the Householder-John theorem without proof.]
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A2.19 B2.19
Part II, 2001 comment(i) Define -step BDF (backward differential formula) methods for the numerical solution of ordinary differential equations and derive explicitly their coefficients.
(ii) Prove that the linear stability domain of the two-step BDF method includes the interval .
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A3.19 B3.20
Part II, 2001 comment(i) The diffusion equation
is discretized by the finite-difference method
where and is a constant. Derive the order of magnitude (as a power of ) of the local error for different choices of .
(ii) Investigate the stability of the above finite-difference method for different values of by the Fourier technique.
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A4.22 B4.20
Part II, 2001 commentWrite an essay on the computation of eigenvalues and eigenvectors of matrices.
-
B1.5
Part II, 2001 commentLet where . Prove that, if is 1-intersecting, then . State an upper bound on that is valid if is -intersecting and is large compared to and .
Let be maximal 1-intersecting; that is, is 1-intersecting but if and then is not 1-intersecting. Show that .
Let be 2 -intersecting. Show that is possible. Can the inequality be strict?
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B2.5
Part II, 2001 commentAs usual, denotes the smallest integer such that every -colouring of yields a monochromatic -subset . Prove that for .
Let have the colex order, and for let ; thus means . Show that if then , and that
Given a red-blue colouring of , the 4 -colouring
is defined as follows:
where . Show that if is monochromatic then is monochromatic, where and .
Deduce that for .
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B4.1
Part II, 2001 commentWrite an essay on extremal graph theory. You should give proofs of at least two major theorems and you should also include a description of alternative proofs or further results.
-
B1.6
Part II, 2001 commentCompute the character table of (begin by listing the conjugacy classes and their orders).
[It is not enough to write down the result; you must justify your answer.]
-
B2.6
Part II, 2001 comment(i) Let be a group, and and finite -sets. Define the permutation representation and compute its character. Show that
is equal to the number of -orbits in .
(ii) Let , and
be the set of 2 -element subsets of . Decompose into irreducibles, and determine the dimension of each irreducible constituent.
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B3.5
Part II, 2001 commentLet , and be the vector space of homogeneous polynomials of degree in the variables and .
(i) Define the action of on , and prove that is an irreducible representation of .
(ii) Decompose into irreducible representations of . Briefly justify your answer.
(iii) acts on the vector space of complex matrices via
Decompose this representation into irreducible representations.
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B4.2
Part II, 2001 commentLet be the Heisenberg group of order . This is the subgroup
of matrices over the finite field ( prime). Let be the subgroup of of such matrices with .
(i) Find all one dimensional representations of .
[You may assume without proof that is equal to the set of matrices in with
(ii) Let be a non-trivial one dimensional representation of , and define a one dimensional representation of by
Show that is irreducible.
(iii) List all the irreducible representations of and explain why your list is complete.
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B1.7
Part II, 2001 commentProve that the Galois group of the polynomial over is of order 6 . By explicitly describing the elements of , show that they have orders 1,2 or 3 . Hence deduce that is isomorphic to .
Why does it follow that is reducible over the finite field , for all primes
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B3.6
Part II, 2001 commentLet be the finite field with elements ( a prime), and let be a finite extension of . Define the Frobenius automorphism , verifying that it is an automorphism of .
Suppose and that is its splitting field over . Why are the zeros of distinct? If is any zero of in , show that . Prove that has at most two zeros in and that . Deduce that the Galois group of over is a cyclic group of order three.
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B4.3
Part II, 2001 commentDefine the concept of separability and normality for algebraic field extensions. Suppose is a simple algebraic extension of , and that denotes the group of -automorphisms of . Prove that , with equality if and only if is normal and separable.
[You may assume that the splitting field of a separable polynomial is normal and separable over .]
Suppose now that is a finite group of automorphisms of a field , and is the fixed subfield. Prove:
(i) is separable.
(ii) and .
(iii) is normal.
[The Primitive Element Theorem for finite separable extensions may be used without proof.]
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B1.8
Part II, 2001 commentDefine an immersion and an embedding of one manifold in another. State a necessary and sufficient condition for an immersion to be an embedding and prove its necessity.
Assuming the existence of "bump functions" on Euclidean spaces, state and prove a version of Whitney's embedding theorem.
Deduce that embeds in .
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B2.7
Part II, 2001 commentState Stokes' Theorem.
Prove that, if is a compact connected manifold and is a surjective chart on , then for any there is such that , where is the unit ball in .
[You may assume that, if with and , then with such that
By considering the -form
on , or otherwise, deduce that .
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B4.4
Part II, 2001 commentDescribe the Mayer-Vietoris exact sequence for forms on a manifold and show how to derive from it the Mayer-Vietoris exact sequence for the de Rham cohomology.
Calculate .
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B2.8
Part II, 2001 commentShow that the fundamental group of the 2-torus is isomorphic to .
Show that an injective continuous map from the circle to itself induces multiplication by on the fundamental group.
Show that there is no retraction from the solid torus to its boundary.
-
B3.7
Part II, 2001 commentWrite down the Mayer-Vietoris sequence and describe all the maps involved.
Use the Mayer-Vietoris sequence to compute the homology of the -sphere for all .
-
B4.5
Part II, 2001 commentWrite an essay on the definition of simplicial homology groups. The essay should include a discussion of orientations, of the action of a simplicial map and a proof of .
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B1.10
Part II, 2001 commentState and prove the Riesz representation theorem for bounded linear functionals on a Hilbert space .
[You may assume, without proof, that , for every closed subspace of .]
Prove that, for every , there is a unique such that for every . Prove that for every .
Define a normal operator . Prove that is normal if and only if for every . Deduce that every point in the spectrum of a normal operator is an approximate eigenvalue of .
[You may assume, without proof, any general criterion for the invertibility of a bounded linear operator on .]
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B1.9
Part II, 2001 commentLet be a number field, where . Let be the (normalized) minimal polynomial of over . Show that the discriminant of is equal to .
Show that is irreducible over Q. Determine and the ring of algebraic integers of , where is a root of .
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B2.9
Part II, 2001 commentDetermine the ideal class group of .
Find all solutions of the diophantine equation
[Minkowski's bound is .]
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B4.6
Part II, 2001 commentFor a prime number , set and .
(a) Show that the (normalized) minimal polynomial of over is equal to
(b) Determine the degrees and .
(c) Show that
(d) Show that .
(e) Show that contains , where .
(f) If are not divisible by , show that lies in .
(g) Show that the ideal is equal to .
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B3.8
Part II, 2001 commentLet be a bounded linear operator on a Hilbert space . Define what it means to say that is (i) compact, and (ii) Fredholm. What is the index, ind , of a Fredholm operator ?
Let be bounded linear operators on . Prove that and are Fredholm if and only if both and are Fredholm. Prove also that if is invertible and is Fredholm then .
Let be a compact linear operator on . Prove that is Fredholm with index zero.
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B4.7
Part II, 2001 commentWrite an essay on the use of Hermite functions in the elementary theory of the Fourier transform on .
[You should assume, without proof, any results that you need concerning the approximation of functions by Hermite functions.]
-
B1.11
Part II, 2001 commentRecall that an automorphism of a Riemann surface is a bijective analytic map onto itself, and that the inverse map is then guaranteed to be analytic.
Let denote the , and let .
(a) Prove that an automorphism with is a Euclidian rotation.
[Hint: Apply the maximum modulus principle to the functions and .]
(b) Prove that a holomorphic map extends to the entire disc, and use this to conclude that any automorphism of is a Euclidean rotation.
[You may use the result stated in part (a).]
(c) Define an analytic map between Riemann surfaces. Show that a continuous map between Riemann surfaces, known to be analytic everywhere except perhaps at a single point , is, in fact, analytic everywhere.
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B3.9
Part II, 2001 commentLet be a nonconstant holomorphic map between compact connected Riemann surfaces. Define the valency of at a point, and the degree of .
Define the genus of a compact connected Riemann surface (assuming the existence of a triangulation).
State the Riemann-Hurwitz theorem. Show that a holomorphic non-constant selfmap of a compact Riemann surface of genus is bijective, with holomorphic inverse. Verify that the Riemann surface in described in the equation is non-singular, and describe its topological type.
[Note: The description can be in the form of a picture or in words. If you apply RiemannHurwitz, explain first how you compactify the surface.]
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B4.8
Part II, 2001 commentLet and be fixed, non-zero complex numbers, with , and let be the lattice they generate in . The series
with the sum taken over all pairs other than , is known to converge to an elliptic function, meaning a meromorphic function satisfying for all . ( is called the Weierstrass function.)
(a) Find the three zeros of modulo , explaining why there are no others.
(b) Show that, for any number , other than the three values and , the equation has exactly two solutions, modulo ; whereas, for each of the specified values, it has a single solution.
[In (a) and (b), you may use, without proof, any known results about valencies and degrees of holomorphic maps between compact Riemann surfaces, provided you state them correctly.]
(c) Prove that every even elliptic function is a rational function of ; that is, there exists a rational function for which .
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B2.10
Part II, 2001 commentLet be the rational map given by : . Determine whether is defined at the following points: .
Let be the curve defined by . Define a bijective morphism . Prove that is not an isomorphism.
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B3.10
Part II, 2001 commentLet be the projective curve (over an algebraically closed field of characteristic zero) defined by the affine equation
Determine the points at infinity of and show that is smooth.
Determine the divisors of the rational functions .
Show that is a regular differential on .
Compute the divisor of . What is the genus of ?
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B4.9
Part II, 2001 commentWrite an essay on curves of genus one (over an algebraically closed field of characteristic zero). Legendre's normal form should not be discussed.
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B1.13
Part II, 2001 commentState and prove Hölder's Inequality.
[Jensen's inequality, and other standard results, may be assumed.]
Let be a sequence of random variables bounded in for some . Prove that is uniformly integrable.
Suppose that for some probability space and some . Show that for all and that is an increasing function of on .
Show further that .
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B2.12
Part II, 2001 comment(a) Let be the Borel -field and let be Lebesgue measure on . What is the distribution of the random variable , where ?
Let be the binary expansion of the point and set , where . Find a random variable independent of such that and are identically distributed and is uniformly distributed on .
(b) Now suppose that on some probability triple and are independent, identicallydistributed random variables such that is uniformly distributed on .
Let be the characteristic function of . Calculate . Show that the distribution of must be the same as the distribution of the random variable in (a).
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B3.12
Part II, 2001 commentState and prove Birkhoff's almost-everywhere ergodic theorem.
[You need not prove convergence in and the maximal ergodic lemma may be assumed provided that it is clearly stated.]
Let be the Borel -field and let be Lebesgue measure on . Give an example of an ergodic measure-preserving map (you need not prove it is ergodic).
Let for . Find (at least for all outside a set of measure zero)
Briefly justify your answer.
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B4.11
Part II, 2001 commentState the first and second Borel-Cantelli Lemmas and the Kolmogorov 0-1 law.
Let be a sequence of independent random variables with distribution given
by
and set .
(a) Show that there exist constants such that , almost surely and almost surely.
(b) Let and , where are independent with
and suppose that .
Use the fact that to show that there exists such that for all sufficiently large .
[You may use the Poisson approximation to the binomial distribution without proof.]
By considering a suitable subsequence of , or otherwise, show that .
(c) Show that . Consider an appropriately chosen sequence of random times , with , for which . Using the fact that the random variables are independent, and by considering the events , or otherwise, show that .
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B2.13
Part II, 2001 commentLet be a Poisson random measure on with constant intensity . For , denote by the line in obtained by rotating the line through an angle about the origin.
Consider the line process .
(i) What is the distribution of the number of lines intersecting the disc ?
(ii) What is the distribution of the distance from the origin to the nearest line?
(iii) What is the distribution of the distance from the origin to the th nearest line?
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B3.13
Part II, 2001 commentConsider an queue with arrival rate and traffic intensity less
than 1. Prove that the moment-generating function of a typical busy period, , satisfies
where is the moment-generating function of a typical service time.
If service times are exponentially distributed with parameter , show that
for all sufficiently small but positive values of .
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B4.12
Part II, 2001 commentDefine a renewal process and a renewal reward process.
State and prove the strong law of large numbers for these processes.
[You may assume the strong law of large numbers for independent, identically-distributed random variables.
State and prove Little's formula.
Customers arrive according to a Poisson process with rate at a single server, but a restricted waiting room causes those who arrive when customers are already present to be lost. Accepted customers have service times which are independent and identicallydistributed with mean and independent of the arrival process. Let be the equilibrium probability that an arriving customer finds customers already present.
Using Little's formula, or otherwise, determine a relationship between and
Part II
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B1.14
Part II, 2001 commentLet be a probability distribution, with . Prove that
All logarithms are to base 2 .
[Hint: To prove (iii), it is convenient to use (i) for and (ii) for .]
Random variables and with values and from finite 'alphabets' and represent the input and output of a transmission channel, with the conditional probability . Let denote the entropy of the conditional distribution , and denote the conditional entropy of given . Define the ideal observer decoding rule as a map such that for all . Show that under this rule the error probability
satisfies , and the expected value satisfies
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B2.14
Part II, 2001 commentA subset of the Hamming space of cardinality and with the minimal (Hamming) distance is called an -code (not necessarily linear). An -code is called maximal if it is not contained in any -code. Prove that an -code is maximal if and only if for any there exists such that . Conclude that if there are or more changes made in a codeword then the new word is closer to some other codeword than to the original one.
Suppose that a maximal -code is used for transmitting information via a binary memoryless channel with the error probability , and the receiver uses the maximum likelihood decoder. Prove that the probability of erroneous decoding, , obeys the bounds
where
is a partial binomial sum and is the integer part.
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B4.13
Part II, 2001 commentState the Kraft inequality. Prove that it gives a necessary and sufficient condition for the existence of a prefix-free code with given codeword lengths.
-
B2.15
Part II, 2001 commentA street trader wishes to dispose of counterfeit Swiss watches. If he offers one for sale at price he will sell it with probability . Here is known and less than 1 . Subsequent to each attempted sale (successful or not) there is a probability that he will be arrested and can make no more sales. His aim is to choose the prices at which he offers the watches so as to maximize the expected values of his sales up until the time he is arrested or has sold all watches.
Let be the maximum expected amount he can obtain when he has watches remaining and has not yet been arrested. Explain why is the solution to
Denote the optimal price by and show that
and that
Show inductively that is a nondecreasing and concave function of .
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B3.14
Part II, 2001 commentA file of is to be transmitted over a communications link. At each time the sender can choose a transmission rate, , within the range Mb per second. The charge for transmitting at rate at time is . The function is fully known at time 0. If it takes a total time to transmit the file then there is a delay cost of , . Thus and are to be chosen to minimize
where and . Quoting and applying appropriate results of Pontryagin's maximum principle show that a property of the optimal policy is that there exists such that if and if .
Show that the optimal and are related by .
Suppose and . For what value of is it optimal to transmit at a constant rate 1 between times and ?
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B4.14
Part II, 2001 commentConsider the scalar system with plant equation and cost
Show from first principles that , where and for
Show that as .
Prove that is minimized by the stationary control, for all .
Consider the stationary policy that has for all . What is the value of under this policy?
Consider the following algorithm, in which steps 1 and 2 are repeated as many times as desired.
- For a given stationary policy , for which for all , determine the value of under this policy as by solving for in
- Now find as the minimizer of
and define as the policy for which for all .
Explain why is guaranteed to be a better policy than .
Let be the stationary policy with . Determine and verify that it minimizes to within of its optimum.
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B1.17
Part II, 2001 commentDefine topological conjugacy and -conjugacy.
Let be real numbers with and let be the maps of to itself given by . For which pairs are and topologically conjugate? Would the answer be the same for -conjugacy? Justify your statements.
-
B3.17
Part II, 2001 commentIf show that for all . Show that has trace 11 and deduce that the subshift map defined by has just two cycles of exact period 5. What are they?
-
B4.17
Part II, 2001 commentDefine the rotation number of an orientation-preserving circle map and the rotation number of a lift of . Prove that and are well-defined. Prove also that is a continuous function of .
State without proof the main consequence of being rational.
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B1.18
Part II, 2001 comment(a) Solve the equation
together with the boundary condition on the -axis:
where is a smooth function. You should discuss the domain on which the solution is smooth. For which functions can the solution be extended to give a smooth solution on the upper half plane ?
(b) Solve the equation
together with the boundary condition on the unit circle:
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B2.17
Part II, 2001 commentDefine the Schwartz space and the corresponding space of tempered distributions .
Use the Fourier transform to give an integral formula for the solution of the equation
for . Prove that your solution lies in . Is your formula the unique solution to in the Schwartz space?
Deduce from this formula an integral expression for the fundamental solution of the operator .
Let be the function:
Using the definition of distributional derivatives verify that this function is a fundamental solution for .
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B3.18
Part II, 2001 commentWrite down a formula for the solution , for and , of the initial value problem for the heat equation:
for a bounded continuous function . State (without proof) a theorem which ensures that this formula is the unique solution in some class of functions (which should be explicitly described).
By writing , or otherwise, solve the initial value problem
for a bounded continuous function and give a class of functions in which your solution is the unique one.
Hence, or otherwise, prove that for all :
and deduce that the solutions and of corresponding to initial values and satisfy, for ,
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B4.18
Part II, 2001 commentWrite an essay on one of the following two topics:
(a) The notion of well-posedness for initial and boundary value problems for differential equations. Your answer should include a definition and give examples and state precise theorems for some specific problems.
(b) The concepts of distribution and tempered distribution and their use in the study of partial differential equations.
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B1.19
Part II, 2001 commentState and prove the convolution theorem for Laplace transforms.
Use the convolution theorem to prove that the Beta function
may be written in terms of the Gamma function as
-
B2.18
Part II, 2001 commentThe Bessel function is defined, for , by
where the path of integration is the Hankel contour and is the principal branch.
Use the method of steepest descent to show that, as ,
You should give a rough sketch of the steepest descent paths.
-
B3.19
Part II, 2001 commentConsider the integral
where is the principal branch and is a positive constant. State the region of the complex -plane in which the integral defines a holomorphic function.
Show how the analytic continuation of this function can be obtained by means of an alternative integral representation using the Hankel contour.
Hence show that the analytic continuation is holomorphic except for simple poles at , and that the residue at is
Part II
-
B4.19
Part II, 2001 commentShow that satisfies the differential equation
and find a second solution, in the form of an integral, for .
Show, by finding the asymptotic behaviour as , that your two solutions are linearly independent.
-
B1.21
Part II, 2001 commentExplain the multipole expansion in electrostatics, and devise formulae for the total charge, dipole moments and quadrupole moments given by a static charge distribution .
A nucleus is modelled as a uniform distribution of charge inside the ellipsoid
The total charge of the nucleus is . What are the dipole moments and quadrupole moments of this distribution?
Describe qualitatively what happens if the nucleus starts to oscillate.
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B2.20
Part II, 2001 commentIn a superconductor, there are superconducting charge carriers with number density , mass and charge . Starting from the quantum mechanical wavefunction (with real and ), construct a formula for the electric current and explain carefully why your result is gauge invariant.
Now show that inside a superconductor a static magnetic field obeys the equation
A superconductor occupies the region , while for there is a vacuum with a constant magnetic field in the direction. Show that the magnetic field cannot penetrate deep into the superconductor.
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B4.21
Part II, 2001 commentThe Liénard-Wiechert potential for a particle of charge , assumed to be moving non-relativistically along the trajectory being the proper time along the trajectory,
Explain how to calculate given and .
Derive Larmor's formula for the rate at which electromagnetic energy is radiated from a particle of charge undergoing an acceleration .
Suppose that one considers the classical non-relativistic hydrogen atom with an electron of mass and charge orbiting a fixed proton of charge , in a circular orbit of radius . What is the total energy of the electron? As the electron is accelerated towards the proton it will radiate, thereby losing energy and causing the orbit to decay. Derive a formula for the lifetime of the orbit.
Part II
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B1.22
Part II, 2001 commentWrite down the first law of thermodynamics in differential form for an infinitesimal reversible change in terms of the increments and , where and are to be defined. Briefly give an interpretation of each term and deduce that
Define the specific heat at constant volume and show that for an adiabatic change
Derive the Maxwell relation
where is temperature and hence show that
An imperfect gas of volume obeys the van der Waals equation of state
where and are non-negative constants. Show that
and deduce that is a function of only. It can further be shown that in this case is independent of . Hence show that
is constant on adiabatic curves.
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B3.22
Part II, 2001 commentA system consists of weakly interacting non-relativistic fermions, each of mass , in a three-dimensional volume, . Derive the Fermi-Dirac distribution
where is the number of particles with energy in and . Explain the physical significance of .
Explain how the constant is determined by the number of particles and write down expressions for and the internal energy in terms of .
Show that, in the limit ,
where .
Show also that in this limit
Deduce that the condition implies that . Discuss briefly whether or not this latter condition is satisfied in a white dwarf star and in a dilute electron gas at room temperature.
Note that .
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B4.23
Part II, 2001 commentGiven that the free energy can be written in terms of the partition function as show that the entropy and internal energy are given by
A system of particles has Hamiltonian where is the set of particle momenta and the set of particle coordinates. Write down the expression for the classical partition function for this system in equilibrium at temperature . In a particular case is given by
Let be a homogeneous function in all the , and in a subset of the . Derive the principle of equipartition for this system.
A system consists of weakly interacting harmonic oscillators each with Hamiltonian
Using equipartition calculate the classical specific heat of the system, . Also calculate the classical entropy .
Write down the expression for the quantum partition function of the system and derive expressions for the specific heat and the entropy in terms of and the parameter . Show for that
where should be calculated. Comment briefly on the physical significance of the constant and why it is non-zero.
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B1.23
Part II, 2001 commentA steady beam of particles, having wavenumber and moving in the direction, scatters on a spherically-symmetric potential. Write down the asymptotic form of the wave function at large .
The incoming wave is written as a partial-wave series
Show that for large
and calculate and for all .
Write down the second-order differential equation satisfied by the . Construct a second linearly-independent solution for each that is singular at and, when it is suitably normalised, has large- behaviour
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B2.22
Part II, 2001 commentA particle of charge moves freely within a cubical box of side . Its initial wavefunction is
A uniform electric field in the direction is switched on for a time . Derive from first principles the probability, correct to order , that after the field has been switched off the wave function will be found to be
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B3.23
Part II, 2001 commentWrite down the commutation relations satisfied by the cartesian components of the total angular momentum operator .
In quantum mechanics an operator is said to be a vector operator if, under rotations, its components transform in the same way as those of the coordinate operator r. Show from first principles that this implies that its cartesian components satisfy the commutation relations
Hence calculate the total angular momentum of the nonvanishing states , where is the vacuum state.
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B4.24
Part II, 2001 commentDerive the Bloch form of the wave function of an electron moving in a onedimensional crystal lattice.
The potential in such an -atom lattice is modelled by
Assuming that is continuous across each lattice site, and applying periodic boundary conditions, derive an equation for the allowed electron energy levels. Show that for suitable values of they have a band structure, and calculate the number of levels in each band when . Verify that when the levels are very close to those corresponding to a solitary atom.
Describe briefly how the band structure in a real 3-dimensional crystal differs from that of this simple model.
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B1.25
Part II, 2001 commentThe energy equation for the motion of a viscous, incompressible fluid states that
Interpret each term in this equation and explain the meaning of the symbols used.
For steady rectilinear flow in a (not necessarily circular) pipe having rigid stationary walls, deduce a relation between the viscous dissipation per unit length of the pipe, the pressure gradient , and the volume flux .
Starting from the Navier-Stokes equations, calculate the velocity field for steady rectilinear flow in a circular pipe of radius . Using the relationship derived above, or otherwise, find in terms of the viscous dissipation per unit length for this flow.
[In cylindrical polar coordinates,
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B2.24
Part II, 2001 commentExplain what is meant by a Stokes flow and show that, in such a flow, in the absence of body forces, , where is the stress tensor.
State and prove the reciprocal theorem for Stokes flow.
When a rigid sphere of radius translates with velocity through unbounded fluid at rest at infinity, it may be shown that the traction per unit area, , exerted by the sphere on the fluid, has the uniform value over the sphere surface. Find the drag on the sphere.
Suppose that the same sphere is free of external forces and is placed with its centre at the origin in an unbounded Stokes flow given in the absence of the sphere as . By applying the reciprocal theorem to the perturbation to the flow generated by the presence of the sphere, and assuming this to tend to zero sufficiently rapidly at infinity, show that the instantaneous velocity of the centre of the sphere is
Part II
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B3.24
Part II, 2001 commentA planar flow of an inviscid, incompressible fluid is everywhere in the -direction and has velocity profile
By examining linear perturbations to the vortex sheet at that have the form , show that
and deduce the temporal stability of the sheet to disturbances of wave number .
Use this result to determine also the spatial growth rate and propagation speed of disturbances of frequency introduced at a fixed spatial position.
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B4.26
Part II, 2001 commentStarting from the steady planar vorticity equation
outline briefly the derivation of the boundary layer equation
explaining the significance of the symbols used.
Viscous fluid occupies the region with rigid stationary walls along for and . There is a line sink at the origin of strength , with . Assuming that vorticity is confined to boundary layers along the rigid walls:
(a) Find the flow outside the boundary layers.
(b) Explain why the boundary layer thickness along the wall is proportional to , and deduce that
(c) Show that the boundary layer equation admits a solution having stream function
Find the equation and boundary conditions satisfied by .
(d) Verify that a solution is
provided that has one of two values to be determined. Should the positive or negative value be chosen?
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B1.26
Part II, 2001 commentDerive Riemann's equations for finite amplitude, one-dimensional sound waves in a perfect gas with ratio of specific heats .
At time the gas is at rest and has uniform density , pressure and sound speed . A piston initially at starts moving backwards at time with displacement , where and are positive constants. Explain briefly how to find the resulting disturbance using a graphical construction in the -plane, and show that prior to any shock forming .
For small amplitude , show that the excess pressure and the excess sound speed are related by
Deduce that the time-averaged pressure on the face of the piston exceeds by
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B2.25
Part II, 2001 commentA semi-infinite elastic medium with shear modulus and shear-wave speed lies in . Above it there is a layer of a second elastic medium with shear modulus and shear-wave speed . The top boundary is stress-free. Consider a monochromatic shear wave propagating at speed with wavenumber in the -direction and with displacements only in the -direction.
Obtain the dispersion relation
Deduce that the modes have a cut-off frequency where they propagate at speed .
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B3.25
Part II, 2001 commentConsider the equation
where and are real constants. Find the dispersion relation for waves of frequency and wavenumber . Find the phase velocity and the group velocity and sketch graphs of these functions.
Multiplying equation by , obtain an equation of the form
where and are expressions involving and its derivatives. Give a physical interpretation of this equation.
Evaluate the time-averaged energy and energy flux of a monochromatic wave , and show that
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B4.27
Part II, 2001 commentDerive the ray-tracing equations governing the evolution of a wave packet in a slowly varying medium, stating the conditions under which the equations are valid.
Consider now a stationary obstacle in a steadily moving homogeneous two-dimensional medium which has the dispersion relation
where is the velocity of the medium. The obstacle generates a steady wave system. Writing , show that the wave satisfies
Show that the group velocity of these waves can be expressed as
Deduce that the waves occupy a wedge of semi-angle about the negative -axis.